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Theorem locfinreflem 32421
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 32420 . . . . 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 496 . 2 (𝜑 → ∃𝑔(𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)))
7 funmpt 6539 . . . . . 6 Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
87a1i 11 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
9 eqid 2736 . . . . . . 7 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
109dmmptss 6193 . . . . . 6 dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ ran 𝑔
11 frn 6675 . . . . . . 7 (𝑔:𝑉𝑈 → ran 𝑔𝑈)
1211ad2antlr 725 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran 𝑔𝑈)
1310, 12sstrid 3955 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈)
14 locfintop 22872 . . . . . . . . . 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 6033 . . . . . . . . . 10 (𝑔 “ {𝑢}) ⊆ dom 𝑔
18 fdm 6677 . . . . . . . . . . 11 (𝑔:𝑉𝑈 → dom 𝑔 = 𝑉)
1918ad3antlr 729 . . . . . . . . . 10 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → dom 𝑔 = 𝑉)
2017, 19sseqtrid 3996 . . . . . . . . 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 3954 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → (𝑔 “ {𝑢}) ⊆ 𝐽)
24 uniopn 22246 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑔 “ {𝑢}) ⊆ 𝐽) → (𝑔 “ {𝑢}) ∈ 𝐽)
2516, 23, 24syl2anc 584 . . . . . . 7 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) → (𝑔 “ {𝑢}) ∈ 𝐽)
2625ralrimiva 3143 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∀𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}) ∈ 𝐽)
279rnmptss 7070 . . . . . 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 2736 . . . . . . . . . 10 𝑉 = 𝑉
30 eqid 2736 . . . . . . . . . 10 𝑈 = 𝑈
3129, 30refbas 22861 . . . . . . . . 9 (𝑉Ref𝑈 𝑈 = 𝑉)
321, 31syl 17 . . . . . . . 8 (𝜑 𝑈 = 𝑉)
3332ad2antrr 724 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑈 = 𝑉)
34 nfv 1917 . . . . . . . . . . . . 13 𝑣(𝜑𝑔:𝑉𝑈)
35 nfra1 3267 . . . . . . . . . . . . 13 𝑣𝑣𝑉 𝑣 ⊆ (𝑔𝑣)
3634, 35nfan 1902 . . . . . . . . . . . 12 𝑣((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))
37 nfre1 3268 . . . . . . . . . . . 12 𝑣𝑣𝑉 𝑥𝑣
3836, 37nfan 1902 . . . . . . . . . . 11 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣)
39 ffn 6668 . . . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . . 14 ((𝑔 Fn 𝑉𝑣𝑉) → (𝑔𝑣) ∈ ran 𝑔)
4340, 41, 42syl2anc 584 . . . . . . . . . . . . 13 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → (𝑔𝑣) ∈ ran 𝑔)
44 ssid 3966 . . . . . . . . . . . . . . 15 𝑣𝑣
4539ad3antlr 729 . . . . . . . . . . . . . . . 16 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) → 𝑔 Fn 𝑉)
46 eqid 2736 . . . . . . . . . . . . . . . . 17 (𝑔𝑣) = (𝑔𝑣)
47 fniniseg 7010 . . . . . . . . . . . . . . . . . 18 (𝑔 Fn 𝑉 → (𝑣 ∈ (𝑔 “ {(𝑔𝑣)}) ↔ (𝑣𝑉 ∧ (𝑔𝑣) = (𝑔𝑣))))
4847biimpar 478 . . . . . . . . . . . . . . . . 17 ((𝑔 Fn 𝑉 ∧ (𝑣𝑉 ∧ (𝑔𝑣) = (𝑔𝑣))) → 𝑣 ∈ (𝑔 “ {(𝑔𝑣)}))
4946, 48mpanr2 702 . . . . . . . . . . . . . . . 16 ((𝑔 Fn 𝑉𝑣𝑉) → 𝑣 ∈ (𝑔 “ {(𝑔𝑣)}))
5045, 49sylancom 588 . . . . . . . . . . . . . . 15 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) → 𝑣 ∈ (𝑔 “ {(𝑔𝑣)}))
51 ssuni 4893 . . . . . . . . . . . . . . 15 ((𝑣𝑣𝑣 ∈ (𝑔 “ {(𝑔𝑣)})) → 𝑣 (𝑔 “ {(𝑔𝑣)}))
5244, 50, 51sylancr 587 . . . . . . . . . . . . . 14 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) → 𝑣 (𝑔 “ {(𝑔𝑣)}))
5352sselda 3944 . . . . . . . . . . . . 13 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑣𝑉) ∧ 𝑥𝑣) → 𝑥 (𝑔 “ {(𝑔𝑣)}))
54 sneq 4596 . . . . . . . . . . . . . . . . 17 (𝑢 = (𝑔𝑣) → {𝑢} = {(𝑔𝑣)})
5554imaeq2d 6013 . . . . . . . . . . . . . . . 16 (𝑢 = (𝑔𝑣) → (𝑔 “ {𝑢}) = (𝑔 “ {(𝑔𝑣)}))
5655unieqd 4879 . . . . . . . . . . . . . . 15 (𝑢 = (𝑔𝑣) → (𝑔 “ {𝑢}) = (𝑔 “ {(𝑔𝑣)}))
5756eleq2d 2823 . . . . . . . . . . . . . 14 (𝑢 = (𝑔𝑣) → (𝑥 (𝑔 “ {𝑢}) ↔ 𝑥 (𝑔 “ {(𝑔𝑣)})))
5857rspcev 3581 . . . . . . . . . . . . 13 (((𝑔𝑣) ∈ ran 𝑔𝑥 (𝑔 “ {(𝑔𝑣)})) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
5943, 53, 58syl2anc 584 . . . . . . . . . . . 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 485 . . . . . . . . . . 11 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣) → ∃𝑣𝑉 𝑥𝑣)
6238, 60, 61r19.29af 3251 . . . . . . . . . 10 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ ∃𝑣𝑉 𝑥𝑣) → ∃𝑢 ∈ ran 𝑔 𝑥 (𝑔 “ {𝑢}))
63 nfv 1917 . . . . . . . . . . . . . 14 𝑣 𝑢 ∈ ran 𝑔
6436, 63nfan 1902 . . . . . . . . . . . . 13 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔)
65 nfv 1917 . . . . . . . . . . . . 13 𝑣 𝑥 (𝑔 “ {𝑢})
6664, 65nfan 1902 . . . . . . . . . . . 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 3945 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → 𝑣𝑉)
70 simpr 485 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) ∧ 𝑥𝑣) → 𝑥𝑣)
71 simpr 485 . . . . . . . . . . . . 13 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) → 𝑥 (𝑔 “ {𝑢}))
72 eluni2 4869 . . . . . . . . . . . . 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 3252 . . . . . . . . . . 11 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑢 ∈ ran 𝑔) ∧ 𝑥 (𝑔 “ {𝑢})) → ∃𝑣𝑉 𝑥𝑣)
7574r19.29an 3155 . . . . . . . . . 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 4869 . . . . . . . . 9 (𝑥 𝑉 ↔ ∃𝑣𝑉 𝑥𝑣)
78 eliun 4958 . . . . . . . . 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 2734 . . . . . . 7 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑉 = 𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
81 dfiun3g 5919 . . . . . . . 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 2780 . . . . . 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 3449 . . . . . . . . . . 11 𝑤 ∈ V
869elrnmpt 5911 . . . . . . . . . . 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 477 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → ∃𝑢 ∈ ran 𝑔 𝑤 = (𝑔 “ {𝑢}))
89 ssrexv 4011 . . . . . . . . 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 1917 . . . . . . . . . 10 𝑢((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣))
92 nfmpt1 5213 . . . . . . . . . . . 12 𝑢(𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9392nfrn 5907 . . . . . . . . . . 11 𝑢ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9493nfcri 2894 . . . . . . . . . 10 𝑢 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9591, 94nfan 1902 . . . . . . . . 9 𝑢(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
96 simpr 485 . . . . . . . . . . 11 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → 𝑤 = (𝑔 “ {𝑢}))
97 nfv 1917 . . . . . . . . . . . . . . . 16 𝑣 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
9836, 97nfan 1902 . . . . . . . . . . . . . . 15 𝑣(((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
99 nfv 1917 . . . . . . . . . . . . . . 15 𝑣 𝑢𝑈
10098, 99nfan 1902 . . . . . . . . . . . . . 14 𝑣((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈)
101 nfv 1917 . . . . . . . . . . . . . 14 𝑣 𝑤 = (𝑔 “ {𝑢})
102100, 101nfan 1902 . . . . . . . . . . . . 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 7010 . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → (𝑣𝑉 ∧ (𝑔𝑣) = 𝑢))
108107simpld 495 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → 𝑣𝑉)
109 rspa 3231 . . . . . . . . . . . . . . . 16 ((∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣) ∧ 𝑣𝑉) → 𝑣 ⊆ (𝑔𝑣))
110103, 108, 109syl2anc 584 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → 𝑣 ⊆ (𝑔𝑣))
111107simprd 496 . . . . . . . . . . . . . . 15 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → (𝑔𝑣) = 𝑢)
112110, 111sseqtrd 3984 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) ∧ 𝑣 ∈ (𝑔 “ {𝑢})) → 𝑣𝑢)
113112ex 413 . . . . . . . . . . . . 13 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → (𝑣 ∈ (𝑔 “ {𝑢}) → 𝑣𝑢))
114102, 113ralrimi 3240 . . . . . . . . . . . 12 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → ∀𝑣 ∈ (𝑔 “ {𝑢})𝑣𝑢)
115 unissb 4900 . . . . . . . . . . . 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 3982 . . . . . . . . . 10 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) ∧ 𝑢𝑈) ∧ 𝑤 = (𝑔 “ {𝑢})) → 𝑤𝑢)
118117exp31 420 . . . . . . . . 9 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))) → (𝑢𝑈 → (𝑤 = (𝑔 “ {𝑢}) → 𝑤𝑢)))
11995, 118reximdai 3244 . . . . . . . 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 3143 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∀𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))∃𝑢𝑈 𝑤𝑢)
122 vex 3449 . . . . . . . . . 10 𝑔 ∈ V
123122rnex 7849 . . . . . . . . 9 ran 𝑔 ∈ V
124123mptex 7173 . . . . . . . 8 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ V
125 rnexg 7841 . . . . . . . 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 2736 . . . . . . . 8 ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))
128127, 30isref 22860 . . . . . . 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 2776 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → 𝑋 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
135 nfv 1917 . . . . . . . . 9 𝑣 𝑥𝑋
13636, 135nfan 1902 . . . . . . . 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 6671 . . . . . . . . . . . . . 14 (𝑔:𝑉𝑈 → Fun 𝑔)
139138ad6antlr 735 . . . . . . . . . . . . 13 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → Fun 𝑔)
140 imafi 9119 . . . . . . . . . . . . 13 ((Fun 𝑔 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) ∈ Fin)
141139, 140sylancom 588 . . . . . . . . . . . 12 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑔 “ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}) ∈ Fin)
142 simp3 1138 . . . . . . . . . . . . . . . . . . 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 4596 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 = 𝑘 → {𝑢} = {𝑘})
144143imaeq2d 6013 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢 = 𝑘 → (𝑔 “ {𝑢}) = (𝑔 “ {𝑘}))
145144unieqd 4879 . . . . . . . . . . . . . . . . . . . . 21 (𝑢 = 𝑘 (𝑔 “ {𝑢}) = (𝑔 “ {𝑘}))
146122cnvex 7862 . . . . . . . . . . . . . . . . . . . . . . 23 𝑔 ∈ V
147 imaexg 7852 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 ∈ V → (𝑔 “ {𝑘}) ∈ V)
148146, 147ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔 “ {𝑘}) ∈ V
149148uniex 7678 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 “ {𝑘}) ∈ V
150145, 9, 149fvmpt 6948 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ran 𝑔 → ((𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))‘𝑘) = (𝑔 “ {𝑘}))
1511503ad2ant2 1134 . . . . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . . . . 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 4171 . . . . . . . . . . . . . . . . 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 3003 . . . . . . . . . . . . . . . 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 7852 . . . . . . . . . . . . . . . . . . . . 21 (𝑔 ∈ V → (𝑔 “ {𝑢}) ∈ V)
157146, 156ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (𝑔 “ {𝑢}) ∈ V
158157uniex 7678 . . . . . . . . . . . . . . . . . . 19 (𝑔 “ {𝑢}) ∈ V
159158, 9fnmpti 6644 . . . . . . . . . . . . . . . . . 18 (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) Fn ran 𝑔
160 dffn4 6762 . . . . . . . . . . . . . . . . . 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 31432 . . . . . . . . . . . . . . 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 4596 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑢 → {𝑘} = {𝑢})
165164imaeq2d 6013 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑢 → (𝑔 “ {𝑘}) = (𝑔 “ {𝑢}))
166165unieqd 4879 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑢 (𝑔 “ {𝑘}) = (𝑔 “ {𝑢}))
167166ineq1d 4171 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑢 → ( (𝑔 “ {𝑘}) ∩ 𝑣) = ( (𝑔 “ {𝑢}) ∩ 𝑣))
168167neeq1d 3003 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑢 → (( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅ ↔ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅))
169168cbvrabv 3417 . . . . . . . . . . . . . . 15 {𝑘 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑘}) ∩ 𝑣) ≠ ∅} = {𝑢 ∈ ran 𝑔 ∣ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅}
170163, 169breqtrdi 5146 . . . . . . . . . . . . . 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 5289 . . . . . . . . . . . . . . 15 {𝑢 ∈ ran 𝑔 ∣ ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢} ∈ V
172 nfv 1917 . . . . . . . . . . . . . . . . . . . . 21 𝑗(((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣)
173 nfrab1 3426 . . . . . . . . . . . . . . . . . . . . . 22 𝑗{𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅}
174173nfel1 2923 . . . . . . . . . . . . . . . . . . . . 21 𝑗{𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin
175172, 174nfan 1902 . . . . . . . . . . . . . . . . . . . 20 𝑗((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin)
176 nfv 1917 . . . . . . . . . . . . . . . . . . . 20 𝑗 𝑢 ∈ ran 𝑔
177175, 176nfan 1902 . . . . . . . . . . . . . . . . . . 19 𝑗(((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔)
178 nfv 1917 . . . . . . . . . . . . . . . . . . 19 𝑗( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅
179177, 178nfan 1902 . . . . . . . . . . . . . . . . . 18 𝑗((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅)
180 nfv 1917 . . . . . . . . . . . . . . . . . . 19 𝑗(𝑔𝑘) = 𝑢
181173, 180nfrexw 3296 . . . . . . . . . . . . . . . . . 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 7010 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔 Fn 𝑉 → (𝑗 ∈ (𝑔 “ {𝑢}) ↔ (𝑗𝑉 ∧ (𝑔𝑗) = 𝑢)))
186185biimpa 477 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔 Fn 𝑉𝑗 ∈ (𝑔 “ {𝑢})) → (𝑗𝑉 ∧ (𝑔𝑗) = 𝑢))
187183, 184, 186syl2anc 584 . . . . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . . . 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 3427 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ↔ (𝑗𝑉 ∧ (𝑗𝑣) ≠ ∅))
191188, 189, 190sylanbrc 583 . . . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . . . . 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 6851 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → ((𝑔𝑘) = 𝑢 ↔ (𝑔𝑗) = 𝑢))
194193rspcev 3581 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∧ (𝑔𝑗) = 𝑢) → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢)
195191, 192, 194syl2anc 584 . . . . . . . . . . . . . . . . . 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 31469 . . . . . . . . . . . . . . . . . . . 20 (( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ ↔ ∃𝑗 ∈ (𝑔 “ {𝑢})(𝑗𝑣) ≠ ∅)
197196biimpi 215 . . . . . . . . . . . . . . . . . . 19 (( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ → ∃𝑗 ∈ (𝑔 “ {𝑢})(𝑗𝑣) ≠ ∅)
198197adantl 482 . . . . . . . . . . . . . . . . . 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 3250 . . . . . . . . . . . . . . . . 17 (((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) ∧ ( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅) → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢)
200199ex 413 . . . . . . . . . . . . . . . 16 ((((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) ∧ 𝑢 ∈ ran 𝑔) → (( (𝑔 “ {𝑢}) ∩ 𝑣) ≠ ∅ → ∃𝑘 ∈ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} (𝑔𝑘) = 𝑢))
201200ss2rabdv 4033 . . . . . . . . . . . . . . 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 8940 . . . . . . . . . . . . . . 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 8947 . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → 𝑔 Fn 𝑉)
207 dffn3 6681 . . . . . . . . . . . . . . 15 (𝑔 Fn 𝑉𝑔:𝑉⟶ran 𝑔)
208207biimpi 215 . . . . . . . . . . . . . 14 (𝑔 Fn 𝑉𝑔:𝑉⟶ran 𝑔)
209 ssrab2 4037 . . . . . . . . . . . . . . 15 {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ⊆ 𝑉
210 fimarab 31559 . . . . . . . . . . . . . . 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 5133 . . . . . . . . . . . 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 9136 . . . . . . . . . . . 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 584 . . . . . . . . . . 11 (((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)
216215ex 413 . . . . . . . . . 10 ((((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) ∧ 𝑥𝑣) → ({𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin → {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
217216imdistanda 572 . . . . . . . . 9 (((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) ∧ 𝑣𝐽) → ((𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin) → (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin)))
218217imp 407 . . . . . . . 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 22868 . . . . . . . . . . . 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 1144 . . . . . . . . . 10 (𝜑 → ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin))
224223r19.21bi 3234 . . . . . . . . 9 ((𝜑𝑥𝑋) → ∃𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin))
225219, 224sylancom 588 . . . . . . . 8 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) → ∃𝑣𝐽 (𝑥𝑣 ∧ {𝑗𝑉 ∣ (𝑗𝑣) ≠ ∅} ∈ Fin))
226136, 137, 218, 225reximd2a 3252 . . . . . . 7 ((((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) ∧ 𝑥𝑋) → ∃𝑣𝐽 (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
227226ralrimiva 3143 . . . . . 6 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∀𝑥𝑋𝑣𝐽 (𝑥𝑣 ∧ {𝑤 ∈ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∣ (𝑤𝑣) ≠ ∅} ∈ Fin))
228220, 127islocfin 22868 . . . . . 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 1343 . . . . 5 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽))
230 funeq 6521 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (Fun 𝑓 ↔ Fun (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))))
231 dmeq 5859 . . . . . . . . 9 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → dom 𝑓 = dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
232231sseq1d 3975 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (dom 𝑓𝑈 ↔ dom (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝑈))
233 rneq 5891 . . . . . . . . 9 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → ran 𝑓 = ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})))
234233sseq1d 3975 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (ran 𝑓𝐽 ↔ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ⊆ 𝐽))
235230, 232, 2343anbi123d 1436 . . . . . . 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 5115 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (ran 𝑓Ref𝑈 ↔ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢}))Ref𝑈))
237233eleq1d 2822 . . . . . . . 8 (𝑓 = (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran (𝑢 ∈ ran 𝑔 (𝑔 “ {𝑢})) ∈ (LocFin‘𝐽)))
238236, 237anbi12d 631 . . . . . . 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 631 . . . . . 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 3565 . . . . 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 1378 . . . 4 (((𝜑𝑔:𝑉𝑈) ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
242241expl 458 . . 3 (𝜑 → ((𝑔:𝑉𝑈 ∧ ∀𝑣𝑉 𝑣 ⊆ (𝑔𝑣)) → ∃𝑓((Fun 𝑓 ∧ dom 𝑓𝑈 ∧ ran 𝑓𝐽) ∧ (ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))))
243242exlimdv 1936 . 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 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cin 3909  wss 3910  c0 4282  {csn 4586   cuni 4865   ciun 4954   class class class wbr 5105  cmpt 5188  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636  Fun wfun 6490   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  cdom 8881  Fincfn 8883  Topctop 22242  Refcref 22853  LocFinclocfin 22855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577  ax-ac2 10399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-fin 8887  df-r1 9700  df-rank 9701  df-card 9875  df-ac 10052  df-top 22243  df-ref 22856  df-locfin 22858
This theorem is referenced by:  locfinref  32422
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