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Theorem locfinreflem 33655
Description: A locally finite refinement of an open cover induces a locally finite open cover with the original index set. This is fact 2 of http://at.yorku.ca/p/a/c/a/02.pdf, it is expressed by exposing a function 𝑓 from the original cover 𝑈, which is taken as the index set. The solution is constructed by building unions, so the same method can be used to prove a similar theorem about closed covers. (Contributed by Thierry Arnoux, 29-Jan-2020.)
Hypotheses
Ref Expression
locfinref.x 𝑋 = 𝐽
locfinref.1 (𝜑𝑈𝐽)
locfinref.2 (𝜑𝑋 = 𝑈)
locfinref.3 (𝜑𝑉𝐽)
locfinref.4 (𝜑𝑉Ref𝑈)
locfinref.5 (𝜑𝑉 ∈ (LocFin‘𝐽))
Assertion
Ref Expression
locfinreflem (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
Distinct variable groups:   𝑓,𝐽   𝑈,𝑓   𝑓,𝑉   𝜑,𝑓
Allowed substitution hint:   𝑋(𝑓)

Proof of Theorem locfinreflem
Dummy variables 𝑔 𝑗 𝑘 𝑢 𝑣 𝑤 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 locfinref.4 . . . 4 (𝜑𝑉Ref𝑈)
2 locfinref.5 . . . . 5 (𝜑𝑉 ∈ (LocFin‘𝐽))
3 reff 33654 . . . . 5 (𝑉 ∈ (LocFin‘𝐽) → (𝑉Ref𝑈 ↔ ( 𝑈 𝑉 ∧ ∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)))))
42, 3syl 17 . . . 4 (𝜑 → (𝑉Ref𝑈 ↔ ( 𝑈 𝑉 ∧ ∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)))))
51, 4mpbid 231 . . 3 (𝜑 → ( 𝑈 𝑉 ∧ ∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))))
65simprd 494 . 2 (𝜑 → ∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)))
7 funmpt 6597 . . . . . 6 Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
87a1i 11 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
9 eqid 2726 . . . . . . 7 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
109dmmptss 6252 . . . . . 6 dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ ran 𝑔
11 frn 6735 . . . . . . 7 (𝑔:𝑉𝑈 → ran 𝑔𝑈)
1211ad2antlr 725 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran 𝑔𝑈)
1310, 12sstrid 3991 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈)
14 locfintop 23516 . . . . . . . . . 10 (𝑉 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
152, 14syl 17 . . . . . . . . 9 (𝜑𝐽 ∈ Top)
1615ad3antrrr 728 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → 𝐽 ∈ Top)
17 cnvimass 6091 . . . . . . . . . 10 (𝑔 “ {𝑢}) ⊆ dom 𝑔
18 fdm 6737 . . . . . . . . . . 11 (𝑔:𝑉𝑈 → dom 𝑔 = 𝑉)
1918ad3antlr 729 . . . . . . . . . 10 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → dom 𝑔 = 𝑉)
2017, 19sseqtrid 4032 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → (𝑔 “ {𝑢}) ⊆ 𝑉)
21 locfinref.3 . . . . . . . . . 10 (𝜑𝑉𝐽)
2221ad3antrrr 728 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → 𝑉𝐽)
2320, 22sstrd 3990 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → (𝑔 “ {𝑢}) ⊆ 𝐽)
24 uniopn 22890 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑔 “ {𝑢}) ⊆ 𝐽) → (𝑔 “ {𝑢}) ∈ 𝐽)
2516, 23, 24syl2anc 582 . . . . . . 7 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → (𝑔 “ {𝑢}) ∈ 𝐽)
2625ralrimiva 3136 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∀𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) ∈ 𝐽)
279rnmptss 7137 . . . . . 6 (∀𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) ∈ 𝐽 → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽)
2826, 27syl 17 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽)
29 eqid 2726 . . . . . . . . . 10 𝑉 = 𝑉
30 eqid 2726 . . . . . . . . . 10 𝑈 = 𝑈
3129, 30refbas 23505 . . . . . . . . 9 (𝑉Ref𝑈 𝑈 = 𝑉)
321, 31syl 17 . . . . . . . 8 (𝜑 𝑈 = 𝑉)
3332ad2antrr 724 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑈 = 𝑉)
34 nfv 1910 . . . . . . . . . . . . 13 𝑣(𝜑𝑔:𝑉𝑈)
35 nfra1 3272 . . . . . . . . . . . . 13 𝑣𝑣𝑉 𝑣 ⊆ (𝑔𝑣)
3634, 35nfan 1895 . . . . . . . . . . . 12 𝑣((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))
37 nfre1 3273 . . . . . . . . . . . 12 𝑣𝑣𝑉 𝑥𝑣
3836, 37nfan 1895 . . . . . . . . . . 11 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣)
39 ffn 6728 . . . . . . . . . . . . . . 15 (𝑔:𝑉𝑈𝑔 Fn 𝑉)
4039ad4antlr 731 . . . . . . . . . . . . . 14 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → 𝑔 Fn 𝑉)
41 simplr 767 . . . . . . . . . . . . . 14 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → 𝑣𝑉)
42 fnfvelrn 7094 . . . . . . . . . . . . . 14 ((𝑔 Fn 𝑉𝑣𝑉) → (𝑔𝑣) ∈ ran 𝑔)
4340, 41, 42syl2anc 582 . . . . . . . . . . . . 13 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → (𝑔𝑣) ∈ ran 𝑔)
44 ssid 4002 . . . . . . . . . . . . . . 15 𝑣𝑣
4539ad3antlr 729 . . . . . . . . . . . . . . . 16 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) → 𝑔 Fn 𝑉)
46 eqid 2726 . . . . . . . . . . . . . . . . 17 (𝑔𝑣) = (𝑔𝑣)
47 fniniseg 7073 . . . . . . . . . . . . . . . . . 18 (𝑔 Fn 𝑉 → (𝑣 ∈ (𝑔 “ {(𝑔𝑣)}) ↔ (𝑣𝑉 ∧ (𝑔𝑣) = (𝑔𝑣))))
4847biimpar 476 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn 𝑉 ∧ (𝑣𝑉 ∧ (𝑔𝑣) = (𝑔𝑣))) → 𝑣 ∈ (𝑔 “ {(𝑔𝑣)}))
4946, 48mpanr2 702 . . . . . . . . . . . . . . . 16 ((𝑔 Fn 𝑉𝑣𝑉) → 𝑣 ∈ (𝑔 “ {(𝑔𝑣)}))
5045, 49sylancom 586 . . . . . . . . . . . . . . 15 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) → 𝑣 ∈ (𝑔 “ {(𝑔𝑣)}))
51 ssuni 4940 . . . . . . . . . . . . . . 15 ((𝑣𝑣𝑣 ∈ (𝑔 “ {(𝑔𝑣)})) → 𝑣 (𝑔 “ {(𝑔𝑣)}))
5244, 50, 51sylancr 585 . . . . . . . . . . . . . 14 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) → 𝑣 (𝑔 “ {(𝑔𝑣)}))
5352sselda 3979 . . . . . . . . . . . . 13 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → 𝑥 (𝑔 “ {(𝑔𝑣)}))
54 sneq 4643 . . . . . . . . . . . . . . . . 17 (𝑢 = (𝑔𝑣) → {𝑢} = {(𝑔𝑣)})
5554imaeq2d 6069 . . . . . . . . . . . . . . . 16 (𝑢 = (𝑔𝑣) → (𝑔 “ {𝑢}) = (𝑔 “ {(𝑔𝑣)}))
5655unieqd 4926 . . . . . . . . . . . . . . 15 (𝑢 = (𝑔𝑣) → (𝑔 “ {𝑢}) = (𝑔 “ {(𝑔𝑣)}))
5756eleq2d 2812 . . . . . . . . . . . . . 14 (𝑢 = (𝑔𝑣) → (𝑥 (𝑔 “ {𝑢}) ↔ 𝑥 (𝑔 “ {(𝑔𝑣)})))
5857rspcev 3608 . . . . . . . . . . . . 13 (((𝑔𝑣) ∈ ran 𝑔𝑥 (𝑔 “ {(𝑔𝑣)})) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
5943, 53, 58syl2anc 582 . . . . . . . . . . . 12 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
6059adantllr 717 . . . . . . . . . . 11 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
61 simpr 483 . . . . . . . . . . 11 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣) → ∃𝑣𝑉 𝑥𝑣)
6238, 60, 61r19.29af 3256 . . . . . . . . . 10 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
63 nfv 1910 . . . . . . . . . . . . . 14 𝑣 𝑢 ∈ ran 𝑔
6436, 63nfan 1895 . . . . . . . . . . . . 13 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔)
65 nfv 1910 . . . . . . . . . . . . 13 𝑣 𝑥 (𝑔 “ {𝑢})
6664, 65nfan 1895 . . . . . . . . . . . 12 𝑣((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢}))
6720ad3antrrr 728 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → (𝑔 “ {𝑢}) ⊆ 𝑉)
68 simplr 767 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → 𝑣 ∈ (𝑔 “ {𝑢}))
6967, 68sseldd 3980 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → 𝑣𝑉)
70 simpr 483 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → 𝑥𝑣)
71 simpr 483 . . . . . . . . . . . . 13 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) → 𝑥 (𝑔 “ {𝑢}))
72 eluni2 4917 . . . . . . . . . . . . 13 (𝑥 (𝑔 “ {𝑢}) ↔ ∃𝑣 ∈ (𝑔 “ {𝑢})𝑥𝑣)
7371, 72sylib 217 . . . . . . . . . . . 12 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) → ∃𝑣 ∈ (𝑔 “ {𝑢})𝑥𝑣)
7466, 69, 70, 73reximd2a 3257 . . . . . . . . . . 11 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) → ∃𝑣𝑉 𝑥𝑣)
7574r19.29an 3148 . . . . . . . . . 10 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢})) → ∃𝑣𝑉 𝑥𝑣)
7662, 75impbida 799 . . . . . . . . 9 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → (∃𝑣𝑉 𝑥𝑣 ↔ ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢})))
77 eluni2 4917 . . . . . . . . 9 (𝑥 𝑉 ↔ ∃𝑣𝑉 𝑥𝑣)
78 eliun 5005 . . . . . . . . 9 (𝑥 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) ↔ ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
7976, 77, 783bitr4g 313 . . . . . . . 8 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → (𝑥 𝑉𝑥 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
8079eqrdv 2724 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑉 = 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
81 dfiun3g 5971 . . . . . . . 8 (∀𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) ∈ 𝐽 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
8226, 81syl 17 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
8333, 80, 823eqtrd 2770 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑈 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
8411ad3antlr 729 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → ran 𝑔𝑈)
85 vex 3466 . . . . . . . . . . 11 𝑤 ∈ V
869elrnmpt 5962 . . . . . . . . . . 11 (𝑤 ∈ V → (𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ↔ ∃𝑢 ∈ ran 𝑔 𝑤 = (𝑔 “ {𝑢})))
8785, 86mp1i 13 . . . . . . . . . 10 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → (𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ↔ ∃𝑢 ∈ ran 𝑔 𝑤 = (𝑔 “ {𝑢})))
8887biimpa 475 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → ∃𝑢 ∈ ran 𝑔 𝑤 = (𝑔 “ {𝑢}))
89 ssrexv 4049 . . . . . . . . 9 (ran 𝑔𝑈 → (∃𝑢 ∈ ran 𝑔 𝑤 = (𝑔 “ {𝑢}) → ∃𝑢𝑈 𝑤 = (𝑔 “ {𝑢})))
9084, 88, 89sylc 65 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → ∃𝑢𝑈 𝑤 = (𝑔 “ {𝑢}))
91 nfv 1910 . . . . . . . . . 10 𝑢((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))
92 nfmpt1 5261 . . . . . . . . . . . 12 𝑢(𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9392nfrn 5958 . . . . . . . . . . 11 𝑢ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9493nfcri 2883 . . . . . . . . . 10 𝑢 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9591, 94nfan 1895 . . . . . . . . 9 𝑢(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
96 simpr 483 . . . . . . . . . . 11 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → 𝑤 = (𝑔 “ {𝑢}))
97 nfv 1910 . . . . . . . . . . . . . . . 16 𝑣 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9836, 97nfan 1895 . . . . . . . . . . . . . . 15 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
99 nfv 1910 . . . . . . . . . . . . . . 15 𝑣 𝑢𝑈
10098, 99nfan 1895 . . . . . . . . . . . . . 14 𝑣((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈)
101 nfv 1910 . . . . . . . . . . . . . 14 𝑣 𝑤 = (𝑔 “ {𝑢})
102100, 101nfan 1895 . . . . . . . . . . . . 13 𝑣(((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢}))
103 simp-5r 784 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))
10439ad5antlr 733 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → 𝑔 Fn 𝑉)
105 fniniseg 7073 . . . . . . . . . . . . . . . . . . 19 (𝑔 Fn 𝑉 → (𝑣 ∈ (𝑔 “ {𝑢}) ↔ (𝑣𝑉 ∧ (𝑔𝑣) = 𝑢)))
106104, 105syl 17 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → (𝑣 ∈ (𝑔 “ {𝑢}) ↔ (𝑣𝑉 ∧ (𝑔𝑣) = 𝑢)))
107106biimpa 475 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → (𝑣𝑉 ∧ (𝑔𝑣) = 𝑢))
108107simpld 493 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → 𝑣𝑉)
109 rspa 3236 . . . . . . . . . . . . . . . 16 ((∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣) ∧ 𝑣𝑉) → 𝑣 ⊆ (𝑔𝑣))
110103, 108, 109syl2anc 582 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → 𝑣 ⊆ (𝑔𝑣))
111107simprd 494 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → (𝑔𝑣) = 𝑢)
112110, 111sseqtrd 4020 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → 𝑣𝑢)
113112ex 411 . . . . . . . . . . . . 13 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → (𝑣 ∈ (𝑔 “ {𝑢}) → 𝑣𝑢))
114102, 113ralrimi 3245 . . . . . . . . . . . 12 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → ∀𝑣 ∈ (𝑔 “ {𝑢})𝑣𝑢)
115 unissb 4947 . . . . . . . . . . . 12 ( (𝑔 “ {𝑢}) ⊆ 𝑢 ↔ ∀𝑣 ∈ (𝑔 “ {𝑢})𝑣𝑢)
116114, 115sylibr 233 . . . . . . . . . . 11 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → (𝑔 “ {𝑢}) ⊆ 𝑢)
11796, 116eqsstrd 4018 . . . . . . . . . 10 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → 𝑤𝑢)
118117exp31 418 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → (𝑢𝑈 → (𝑤 = (𝑔 “ {𝑢}) → 𝑤𝑢)))
11995, 118reximdai 3249 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → (∃𝑢𝑈 𝑤 = (𝑔 “ {𝑢}) → ∃𝑢𝑈 𝑤𝑢))
12090, 119mpd 15 . . . . . . 7 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → ∃𝑢𝑈 𝑤𝑢)
121120ralrimiva 3136 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∀𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))∃𝑢𝑈 𝑤𝑢)
122 vex 3466 . . . . . . . . . 10 𝑔 ∈ V
123122rnex 7923 . . . . . . . . 9 ran 𝑔 ∈ V
124123mptex 7240 . . . . . . . 8 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V
125 rnexg 7915 . . . . . . . 8 ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V)
126124, 125mp1i 13 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V)
127 eqid 2726 . . . . . . . 8 ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
128127, 30isref 23504 . . . . . . 7 (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V → (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ↔ ( 𝑈 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ ∀𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))∃𝑢𝑈 𝑤𝑢)))
129126, 128syl 17 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ↔ ( 𝑈 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ ∀𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))∃𝑢𝑈 𝑤𝑢)))
13083, 121, 129mpbir2and 711 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈)
13115ad2antrr 724 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝐽 ∈ Top)
132 locfinref.2 . . . . . . . 8 (𝜑𝑋 = 𝑈)
133132ad2antrr 724 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑋 = 𝑈)
134133, 83eqtrd 2766 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑋 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
135 nfv 1910 . . . . . . . . 9 𝑣 𝑥𝑋
13636, 135nfan 1895 . . . . . . . 8 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋)
137 simplr 767 . . . . . . . 8 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)) → 𝑣𝐽)
138 ffun 6731 . . . . . . . . . . . . . 14 (𝑔:𝑉𝑈 → Fun 𝑔)
139138ad6antlr 735 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → Fun 𝑔)
140 imafi 9355 . . . . . . . . . . . . 13 ((Fun 𝑔 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) ∈ Fin)
141139, 140sylancom 586 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) ∈ Fin)
142 simp3 1135 . . . . . . . . . . . . . . . . . . 19 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → 𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘))
143 sneq 4643 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑘 → {𝑢} = {𝑘})
144143imaeq2d 6069 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑘 → (𝑔 “ {𝑢}) = (𝑔 “ {𝑘}))
145144unieqd 4926 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑘 (𝑔 “ {𝑢}) = (𝑔 “ {𝑘}))
146122cnvex 7938 . . . . . . . . . . . . . . . . . . . . . . 23 𝑔 ∈ V
147 imaexg 7926 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 ∈ V → (𝑔 “ {𝑘}) ∈ V)
148146, 147ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 “ {𝑘}) ∈ V
149148uniex 7752 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 “ {𝑘}) ∈ V
150145, 9, 149fvmpt 7009 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ran 𝑔 → ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘) = (𝑔 “ {𝑘}))
1511503ad2ant2 1131 . . . . . . . . . . . . . . . . . . 19 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘) = (𝑔 “ {𝑘}))
152142, 151eqtrd 2766 . . . . . . . . . . . . . . . . . 18 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → 𝑤 = (𝑔 “ {𝑘}))
153152ineq1d 4212 . . . . . . . . . . . . . . . . 17 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → (𝑤𝑣) = ( (𝑔 “ {𝑘}) ∩ 𝑣))
154153neeq1d 2990 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑘 ∈ ran 𝑔𝑤 = ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘)) → ((𝑤𝑣) ≠ ∅ ↔ ( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅))
155123a1i 11 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → ran 𝑔 ∈ V)
156 imaexg 7926 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 ∈ V → (𝑔 “ {𝑢}) ∈ V)
157146, 156ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (𝑔 “ {𝑢}) ∈ V
158157uniex 7752 . . . . . . . . . . . . . . . . . . 19 (𝑔 “ {𝑢}) ∈ V
159158, 9fnmpti 6704 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) Fn ran 𝑔
160 dffn4 6821 . . . . . . . . . . . . . . . . . 18 ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) Fn ran 𝑔 ↔ (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})):ran 𝑔onto→ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
161159, 160mpbi 229 . . . . . . . . . . . . . . . . 17 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})):ran 𝑔onto→ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
162161a1i 11 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})):ran 𝑔onto→ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
163154, 155, 162rabfodom 32431 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑘 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅})
164 sneq 4643 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑢 → {𝑘} = {𝑢})
165164imaeq2d 6069 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑢 → (𝑔 “ {𝑘}) = (𝑔 “ {𝑢}))
166165unieqd 4926 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑢 (𝑔 “ {𝑘}) = (𝑔 “ {𝑢}))
167166ineq1d 4212 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑢 → ( (𝑔 “ {𝑘}) ∩ 𝑣) = ( (𝑔 “ {𝑢}) ∩ 𝑣))
168167neeq1d 2990 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑢 → (( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅ ↔ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅))
169168cbvrabv 3430 . . . . . . . . . . . . . . 15 {𝑘 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅} = {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅}
170163, 169breqtrdi 5194 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅})
171123rabex 5339 . . . . . . . . . . . . . . 15 {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢} ∈ V
172 nfv 1910 . . . . . . . . . . . . . . . . . . . . 21 𝑗(((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣)
173 nfrab1 3439 . . . . . . . . . . . . . . . . . . . . . 22 𝑗{𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}
174173nfel1 2909 . . . . . . . . . . . . . . . . . . . . 21 𝑗{𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin
175172, 174nfan 1895 . . . . . . . . . . . . . . . . . . . 20 𝑗((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)
176 nfv 1910 . . . . . . . . . . . . . . . . . . . 20 𝑗 𝑢 ∈ ran 𝑔
177175, 176nfan 1895 . . . . . . . . . . . . . . . . . . 19 𝑗(((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔)
178 nfv 1910 . . . . . . . . . . . . . . . . . . 19 𝑗( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅
179177, 178nfan 1895 . . . . . . . . . . . . . . . . . 18 𝑗((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅)
180 nfv 1910 . . . . . . . . . . . . . . . . . . 19 𝑗(𝑔𝑘) = 𝑢
181173, 180nfrexw 3301 . . . . . . . . . . . . . . . . . 18 𝑗𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢
18239ad5antlr 733 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) → 𝑔 Fn 𝑉)
183182ad5antr 732 . . . . . . . . . . . . . . . . . . . . . 22 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → 𝑔 Fn 𝑉)
184 simplr 767 . . . . . . . . . . . . . . . . . . . . . 22 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → 𝑗 ∈ (𝑔 “ {𝑢}))
185 fniniseg 7073 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 Fn 𝑉 → (𝑗 ∈ (𝑔 “ {𝑢}) ↔ (𝑗𝑉 ∧ (𝑔𝑗) = 𝑢)))
186185biimpa 475 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 Fn 𝑉𝑗 ∈ (𝑔 “ {𝑢})) → (𝑗𝑉 ∧ (𝑔𝑗) = 𝑢))
187183, 184, 186syl2anc 582 . . . . . . . . . . . . . . . . . . . . 21 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → (𝑗𝑉 ∧ (𝑔𝑗) = 𝑢))
188187simpld 493 . . . . . . . . . . . . . . . . . . . 20 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → 𝑗𝑉)
189 simpr 483 . . . . . . . . . . . . . . . . . . . 20 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → (𝑗𝑣) ≠ ∅)
190 rabid 3440 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ↔ (𝑗𝑉 ∧ (𝑗𝑣) ≠ ∅))
191188, 189, 190sylanbrc 581 . . . . . . . . . . . . . . . . . . 19 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → 𝑗 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅})
192187simprd 494 . . . . . . . . . . . . . . . . . . 19 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → (𝑔𝑗) = 𝑢)
193 fveqeq2 6910 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → ((𝑔𝑘) = 𝑢 ↔ (𝑔𝑗) = 𝑢))
194193rspcev 3608 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∧ (𝑔𝑗) = 𝑢) → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢)
195191, 192, 194syl2anc 582 . . . . . . . . . . . . . . . . . 18 (((((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) ∧ 𝑗 ∈ (𝑔 “ {𝑢})) ∧ (𝑗𝑣) ≠ ∅) → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢)
196 uniinn0 32471 . . . . . . . . . . . . . . . . . . . 20 (( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ ↔ ∃𝑗 ∈ (𝑔 “ {𝑢})(𝑗𝑣) ≠ ∅)
197196biimpi 215 . . . . . . . . . . . . . . . . . . 19 (( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ → ∃𝑗 ∈ (𝑔 “ {𝑢})(𝑗𝑣) ≠ ∅)
198197adantl 480 . . . . . . . . . . . . . . . . . 18 (((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) → ∃𝑗 ∈ (𝑔 “ {𝑢})(𝑗𝑣) ≠ ∅)
199179, 181, 195, 198r19.29af2 3255 . . . . . . . . . . . . . . . . 17 (((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢)
200199ex 411 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) → (( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢))
201200ss2rabdv 4072 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ⊆ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
202 ssdomg 9031 . . . . . . . . . . . . . . 15 ({𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢} ∈ V → ({𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ⊆ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢} → {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢}))
203171, 201, 202mpsyl 68 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
204 domtr 9038 . . . . . . . . . . . . . 14 (({𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ∧ {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢}) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
205170, 203, 204syl2anc 582 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
206182adantr 479 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → 𝑔 Fn 𝑉)
207 dffn3 6740 . . . . . . . . . . . . . . 15 (𝑔 Fn 𝑉𝑔:𝑉⟶ran 𝑔)
208207biimpi 215 . . . . . . . . . . . . . 14 (𝑔 Fn 𝑉𝑔:𝑉⟶ran 𝑔)
209 ssrab2 4076 . . . . . . . . . . . . . . 15 {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ⊆ 𝑉
210 fimarab 32560 . . . . . . . . . . . . . . 15 ((𝑔:𝑉⟶ran 𝑔 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ⊆ 𝑉) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) = {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
211209, 210mpan2 689 . . . . . . . . . . . . . 14 (𝑔:𝑉⟶ran 𝑔 → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) = {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
212206, 208, 2113syl 18 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) = {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢})
213205, 212breqtrrd 5181 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}))
214 domfi 9226 . . . . . . . . . . . 12 (((𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) ∈ Fin ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ≼ (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅})) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)
215141, 213, 214syl2anc 582 . . . . . . . . . . 11 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)
216215ex 411 . . . . . . . . . 10 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) → ({𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
217216imdistanda 570 . . . . . . . . 9 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) → ((𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)))
218217imp 405 . . . . . . . 8 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)) → (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
219 simplll 773 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) → 𝜑)
220 locfinref.x . . . . . . . . . . . . 13 𝑋 = 𝐽
221220, 29islocfin 23512 . . . . . . . . . . . 12 (𝑉 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑉 ∧ ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)))
2222, 221sylib 217 . . . . . . . . . . 11 (𝜑 → (𝐽 ∈ Top ∧ 𝑋 = 𝑉 ∧ ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)))
223222simp3d 1141 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin))
224223r19.21bi 3239 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∃𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin))
225219, 224sylancom 586 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) → ∃𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin))
226136, 137, 218, 225reximd2a 3257 . . . . . . 7 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) → ∃𝑣𝐽 (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
227226ralrimiva 3136 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
228220, 127islocfin 23512 . . . . . 6 (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)))
229131, 134, 227, 228syl3anbrc 1340 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽))
230 funeq 6579 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (Fun 𝑓 ↔ Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))))
231 dmeq 5910 . . . . . . . . 9 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → dom 𝑓 = dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
232231sseq1d 4011 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (dom 𝑓𝑈 ↔ dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈))
233 rneq 5942 . . . . . . . . 9 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → ran 𝑓 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
234233sseq1d 4011 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (ran 𝑓𝐽 ↔ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽))
235230, 232, 2343anbi123d 1433 . . . . . . 7 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → ((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ↔ (Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽)))
236233breq1d 5163 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (ran 𝑓Ref𝑈 ↔ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈))
237233eleq1d 2811 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽)))
238236, 237anbi12d 630 . . . . . . 7 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → ((ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)) ↔ (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽))))
239235, 238anbi12d 630 . . . . . 6 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))) ↔ ((Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽) ∧ (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽)))))
240124, 239spcev 3592 . . . . 5 (((Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∧ dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽) ∧ (ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈 ∧ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽))) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
2418, 13, 28, 130, 229, 240syl32anc 1375 . . . 4 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
242241expl 456 . . 3 (𝜑 → ((𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))))
243242exlimdv 1929 . 2 (𝜑 → (∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))))
2446, 243mpd 15 1 (𝜑 → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wex 1774  wcel 2099  wne 2930  wral 3051  wrex 3060  {crab 3419  Vcvv 3462  cin 3946  wss 3947  c0 4325  {csn 4633   cuni 4913   ciun 5001   class class class wbr 5153  cmpt 5236  ccnv 5681  dom cdm 5682  ran crn 5683  cima 5685  Fun wfun 6548   Fn wfn 6549  wf 6550  ontowfo 6552  cfv 6554  cdom 8972  Fincfn 8974  Topctop 22886  Refcref 23497  LocFinclocfin 23499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-reg 9635  ax-inf2 9684  ax-ac2 10506
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-en 8975  df-dom 8976  df-fin 8978  df-r1 9807  df-rank 9808  df-card 9982  df-ac 10159  df-top 22887  df-ref 23500  df-locfin 23502
This theorem is referenced by:  locfinref  33656
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