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Theorem locfinref 29042
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. (Contributed by Thierry Arnoux, 31-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
locfinref (𝜑 → ∃𝑓(𝑓:𝑈𝐽 ∧ 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 locfinref
Dummy variables 𝑔 𝑥 𝑛 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f0 5984 . . . 4 ∅:∅⟶𝐽
2 simpr 475 . . . . 5 ((𝜑𝑈 = ∅) → 𝑈 = ∅)
32feq2d 5930 . . . 4 ((𝜑𝑈 = ∅) → (∅:𝑈𝐽 ↔ ∅:∅⟶𝐽))
41, 3mpbiri 246 . . 3 ((𝜑𝑈 = ∅) → ∅:𝑈𝐽)
5 rn0 5285 . . . . 5 ran ∅ = ∅
6 0ex 4713 . . . . . 6 ∅ ∈ V
7 refref 21068 . . . . . 6 (∅ ∈ V → ∅Ref∅)
86, 7ax-mp 5 . . . . 5 ∅Ref∅
95, 8eqbrtri 4598 . . . 4 ran ∅Ref∅
109, 2syl5breqr 4615 . . 3 ((𝜑𝑈 = ∅) → ran ∅Ref𝑈)
11 sn0top 20555 . . . . . 6 {∅} ∈ Top
1211a1i 11 . . . . 5 ((𝜑𝑈 = ∅) → {∅} ∈ Top)
13 eqidd 2610 . . . . 5 ((𝜑𝑈 = ∅) → ∅ = ∅)
14 ral0 4027 . . . . . 6 𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)
1514a1i 11 . . . . 5 ((𝜑𝑈 = ∅) → ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
166unisn 4381 . . . . . . 7 {∅} = ∅
1716eqcomi 2618 . . . . . 6 ∅ = {∅}
185unieqi 4375 . . . . . . 7 ran ∅ =
19 uni0 4395 . . . . . . 7 ∅ = ∅
2018, 19eqtr2i 2632 . . . . . 6 ∅ = ran ∅
2117, 20islocfin 21072 . . . . 5 (ran ∅ ∈ (LocFin‘{∅}) ↔ ({∅} ∈ Top ∧ ∅ = ∅ ∧ ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
2212, 13, 15, 21syl3anbrc 1238 . . . 4 ((𝜑𝑈 = ∅) → ran ∅ ∈ (LocFin‘{∅}))
23 locfinref.2 . . . . . . . . 9 (𝜑𝑋 = 𝑈)
2423adantr 479 . . . . . . . 8 ((𝜑𝑈 = ∅) → 𝑋 = 𝑈)
252unieqd 4376 . . . . . . . 8 ((𝜑𝑈 = ∅) → 𝑈 = ∅)
2624, 25eqtrd 2643 . . . . . . 7 ((𝜑𝑈 = ∅) → 𝑋 = ∅)
27 locfinref.x . . . . . . 7 𝑋 = 𝐽
2826, 27, 193eqtr3g 2666 . . . . . 6 ((𝜑𝑈 = ∅) → 𝐽 = ∅)
29 locfinref.5 . . . . . . . 8 (𝜑𝑉 ∈ (LocFin‘𝐽))
30 locfintop 21076 . . . . . . . 8 (𝑉 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
31 0top 20540 . . . . . . . 8 (𝐽 ∈ Top → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
3229, 30, 313syl 18 . . . . . . 7 (𝜑 → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
3332adantr 479 . . . . . 6 ((𝜑𝑈 = ∅) → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
3428, 33mpbid 220 . . . . 5 ((𝜑𝑈 = ∅) → 𝐽 = {∅})
3534fveq2d 6092 . . . 4 ((𝜑𝑈 = ∅) → (LocFin‘𝐽) = (LocFin‘{∅}))
3622, 35eleqtrrd 2690 . . 3 ((𝜑𝑈 = ∅) → ran ∅ ∈ (LocFin‘𝐽))
37 feq1 5925 . . . . 5 (𝑓 = ∅ → (𝑓:𝑈𝐽 ↔ ∅:𝑈𝐽))
38 rneq 5259 . . . . . 6 (𝑓 = ∅ → ran 𝑓 = ran ∅)
3938breq1d 4587 . . . . 5 (𝑓 = ∅ → (ran 𝑓Ref𝑈 ↔ ran ∅Ref𝑈))
4038eleq1d 2671 . . . . 5 (𝑓 = ∅ → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran ∅ ∈ (LocFin‘𝐽)))
4137, 39, 403anbi123d 1390 . . . 4 (𝑓 = ∅ → ((𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)) ↔ (∅:𝑈𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽))))
426, 41spcev 3272 . . 3 ((∅:𝑈𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽)) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
434, 10, 36, 42syl3anc 1317 . 2 ((𝜑𝑈 = ∅) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
44 locfinref.1 . . . . 5 (𝜑𝑈𝐽)
45 locfinref.3 . . . . 5 (𝜑𝑉𝐽)
46 locfinref.4 . . . . 5 (𝜑𝑉Ref𝑈)
4727, 44, 23, 45, 46, 29locfinreflem 29041 . . . 4 (𝜑 → ∃𝑔((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽))))
4847adantr 479 . . 3 ((𝜑𝑈 ≠ ∅) → ∃𝑔((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽))))
49 simpl 471 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (𝜑𝑈 ≠ ∅))
50 simprl1 1098 . . . . . . . 8 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → Fun 𝑔)
51 fdmrn 5963 . . . . . . . 8 (Fun 𝑔𝑔:dom 𝑔⟶ran 𝑔)
5250, 51sylib 206 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑔:dom 𝑔⟶ran 𝑔)
53 simprl3 1100 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔𝐽)
5452, 53fssd 5956 . . . . . 6 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑔:dom 𝑔𝐽)
55 fconstg 5990 . . . . . . . 8 (∅ ∈ V → ((𝑈 ∖ dom 𝑔) × {∅}):(𝑈 ∖ dom 𝑔)⟶{∅})
566, 55mp1i 13 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ((𝑈 ∖ dom 𝑔) × {∅}):(𝑈 ∖ dom 𝑔)⟶{∅})
57 0opn 20476 . . . . . . . . . 10 (𝐽 ∈ Top → ∅ ∈ 𝐽)
5829, 30, 573syl 18 . . . . . . . . 9 (𝜑 → ∅ ∈ 𝐽)
5958ad2antrr 757 . . . . . . . 8 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ∅ ∈ 𝐽)
6059snssd 4280 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → {∅} ⊆ 𝐽)
6156, 60fssd 5956 . . . . . 6 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ((𝑈 ∖ dom 𝑔) × {∅}):(𝑈 ∖ dom 𝑔)⟶𝐽)
62 disjdif 3991 . . . . . . 7 (dom 𝑔 ∩ (𝑈 ∖ dom 𝑔)) = ∅
6362a1i 11 . . . . . 6 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (dom 𝑔 ∩ (𝑈 ∖ dom 𝑔)) = ∅)
64 fun2 5966 . . . . . 6 (((𝑔:dom 𝑔𝐽 ∧ ((𝑈 ∖ dom 𝑔) × {∅}):(𝑈 ∖ dom 𝑔)⟶𝐽) ∧ (dom 𝑔 ∩ (𝑈 ∖ dom 𝑔)) = ∅) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):(dom 𝑔 ∪ (𝑈 ∖ dom 𝑔))⟶𝐽)
6554, 61, 63, 64syl21anc 1316 . . . . 5 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):(dom 𝑔 ∪ (𝑈 ∖ dom 𝑔))⟶𝐽)
66 simprl2 1099 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → dom 𝑔𝑈)
67 undif 4000 . . . . . . 7 (dom 𝑔𝑈 ↔ (dom 𝑔 ∪ (𝑈 ∖ dom 𝑔)) = 𝑈)
6866, 67sylib 206 . . . . . 6 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (dom 𝑔 ∪ (𝑈 ∖ dom 𝑔)) = 𝑈)
6968feq2d 5930 . . . . 5 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ((𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):(dom 𝑔 ∪ (𝑈 ∖ dom 𝑔))⟶𝐽 ↔ (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈𝐽))
7065, 69mpbid 220 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈𝐽)
71 simpr 475 . . . . . 6 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔)
72 simprrl 799 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔Ref𝑈)
7372adantr 479 . . . . . 6 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔) → ran 𝑔Ref𝑈)
7471, 73eqbrtrd 4599 . . . . 5 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈)
75 simpr 475 . . . . . 6 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
7649simprd 477 . . . . . . . 8 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑈 ≠ ∅)
77 refun0 21070 . . . . . . . 8 ((ran 𝑔Ref𝑈𝑈 ≠ ∅) → (ran 𝑔 ∪ {∅})Ref𝑈)
7872, 76, 77syl2anc 690 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (ran 𝑔 ∪ {∅})Ref𝑈)
7978adantr 479 . . . . . 6 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → (ran 𝑔 ∪ {∅})Ref𝑈)
8075, 79eqbrtrd 4599 . . . . 5 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈)
81 rnxpss 5471 . . . . . . 7 ran ((𝑈 ∖ dom 𝑔) × {∅}) ⊆ {∅}
82 sssn 4295 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) ⊆ {∅} ↔ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅}))
8381, 82mpbi 218 . . . . . 6 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅})
84 rnun 5446 . . . . . . . . 9 ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅}))
85 uneq2 3722 . . . . . . . . 9 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
8684, 85syl5eq 2655 . . . . . . . 8 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
87 un0 3918 . . . . . . . 8 (ran 𝑔 ∪ ∅) = ran 𝑔
8886, 87syl6eq 2659 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔)
89 uneq2 3722 . . . . . . . 8 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
9084, 89syl5eq 2655 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
9188, 90orim12i 536 . . . . . 6 ((ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅}) → (ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔 ∨ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})))
9283, 91mp1i 13 . . . . 5 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔 ∨ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})))
9374, 80, 92mpjaodan 822 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈)
94 simprrr 800 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔 ∈ (LocFin‘𝐽))
9594adantr 479 . . . . . 6 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔) → ran 𝑔 ∈ (LocFin‘𝐽))
9671, 95eqeltrd 2687 . . . . 5 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))
9794adantr 479 . . . . . . 7 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → ran 𝑔 ∈ (LocFin‘𝐽))
98 snfi 7900 . . . . . . . 8 {∅} ∈ Fin
9998a1i 11 . . . . . . 7 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → {∅} ∈ Fin)
10059adantr 479 . . . . . . . . 9 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → ∅ ∈ 𝐽)
101100snssd 4280 . . . . . . . 8 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → {∅} ⊆ 𝐽)
102101unissd 4392 . . . . . . 7 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → {∅} ⊆ 𝐽)
103 lfinun 21080 . . . . . . 7 ((ran 𝑔 ∈ (LocFin‘𝐽) ∧ {∅} ∈ Fin ∧ {∅} ⊆ 𝐽) → (ran 𝑔 ∪ {∅}) ∈ (LocFin‘𝐽))
10497, 99, 102, 103syl3anc 1317 . . . . . 6 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → (ran 𝑔 ∪ {∅}) ∈ (LocFin‘𝐽))
10575, 104eqeltrd 2687 . . . . 5 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))
10696, 105, 92mpjaodan 822 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))
107 refrel 21063 . . . . . . . . 9 Rel Ref
108107brrelex2i 5073 . . . . . . . 8 (𝑉Ref𝑈𝑈 ∈ V)
109 difexg 4730 . . . . . . . 8 (𝑈 ∈ V → (𝑈 ∖ dom 𝑔) ∈ V)
11046, 108, 1093syl 18 . . . . . . 7 (𝜑 → (𝑈 ∖ dom 𝑔) ∈ V)
111110adantr 479 . . . . . 6 ((𝜑𝑈 ≠ ∅) → (𝑈 ∖ dom 𝑔) ∈ V)
112 p0ex 4774 . . . . . . 7 {∅} ∈ V
113 xpexg 6835 . . . . . . 7 (((𝑈 ∖ dom 𝑔) ∈ V ∧ {∅} ∈ V) → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
114112, 113mpan2 702 . . . . . 6 ((𝑈 ∖ dom 𝑔) ∈ V → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
115 vex 3175 . . . . . . 7 𝑔 ∈ V
116 unexg 6834 . . . . . . 7 ((𝑔 ∈ V ∧ ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V)
117115, 116mpan 701 . . . . . 6 (((𝑈 ∖ dom 𝑔) × {∅}) ∈ V → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V)
118 feq1 5925 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (𝑓:𝑈𝐽 ↔ (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈𝐽))
119 rneq 5259 . . . . . . . . 9 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → ran 𝑓 = ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})))
120119breq1d 4587 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓Ref𝑈 ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈))
121119eleq1d 2671 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽)))
122118, 120, 1213anbi123d 1390 . . . . . . 7 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → ((𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)) ↔ ((𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈𝐽 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))))
123122spcegv 3266 . . . . . 6 ((𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V → (((𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈𝐽 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽)) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
124111, 114, 117, 1234syl 19 . . . . 5 ((𝜑𝑈 ≠ ∅) → (((𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈𝐽 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽)) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽))))
125124imp 443 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈𝐽 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈 ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
12649, 70, 93, 106, 125syl13anc 1319 . . 3 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
12748, 126exlimddv 1849 . 2 ((𝜑𝑈 ≠ ∅) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
12843, 127pm2.61dane 2868 1 (𝜑 → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  wne 2779  wral 2895  wrex 2896  {crab 2899  Vcvv 3172  cdif 3536  cun 3537  cin 3538  wss 3539  c0 3873  {csn 4124   cuni 4366   class class class wbr 4577   × cxp 5026  dom cdm 5028  ran crn 5029  Fun wfun 5784  wf 5786  cfv 5790  Fincfn 7818  Topctop 20459  Refcref 21057  LocFinclocfin 21059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-reg 8357  ax-inf2 8398  ax-ac2 9145
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-fin 7822  df-r1 8487  df-rank 8488  df-card 8625  df-ac 8799  df-top 20463  df-topon 20465  df-ref 21060  df-locfin 21062
This theorem is referenced by:  pcmplfinf  29062
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