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Theorem locfinref 30036
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 6124 . . . 4 ∅:∅⟶𝐽
2 simpr 476 . . . . 5 ((𝜑𝑈 = ∅) → 𝑈 = ∅)
32feq2d 6069 . . . 4 ((𝜑𝑈 = ∅) → (∅:𝑈𝐽 ↔ ∅:∅⟶𝐽))
41, 3mpbiri 248 . . 3 ((𝜑𝑈 = ∅) → ∅:𝑈𝐽)
5 rn0 5409 . . . . 5 ran ∅ = ∅
6 0ex 4823 . . . . . 6 ∅ ∈ V
7 refref 21364 . . . . . 6 (∅ ∈ V → ∅Ref∅)
86, 7ax-mp 5 . . . . 5 ∅Ref∅
95, 8eqbrtri 4706 . . . 4 ran ∅Ref∅
109, 2syl5breqr 4723 . . 3 ((𝜑𝑈 = ∅) → ran ∅Ref𝑈)
11 sn0top 20851 . . . . . 6 {∅} ∈ Top
1211a1i 11 . . . . 5 ((𝜑𝑈 = ∅) → {∅} ∈ Top)
13 eqidd 2652 . . . . 5 ((𝜑𝑈 = ∅) → ∅ = ∅)
14 ral0 4109 . . . . . 6 𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)
1514a1i 11 . . . . 5 ((𝜑𝑈 = ∅) → ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
166unisn 4483 . . . . . . 7 {∅} = ∅
1716eqcomi 2660 . . . . . 6 ∅ = {∅}
185unieqi 4477 . . . . . . 7 ran ∅ =
19 uni0 4497 . . . . . . 7 ∅ = ∅
2018, 19eqtr2i 2674 . . . . . 6 ∅ = ran ∅
2117, 20islocfin 21368 . . . . 5 (ran ∅ ∈ (LocFin‘{∅}) ↔ ({∅} ∈ Top ∧ ∅ = ∅ ∧ ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
2212, 13, 15, 21syl3anbrc 1265 . . . 4 ((𝜑𝑈 = ∅) → ran ∅ ∈ (LocFin‘{∅}))
23 locfinref.2 . . . . . . . . 9 (𝜑𝑋 = 𝑈)
2423adantr 480 . . . . . . . 8 ((𝜑𝑈 = ∅) → 𝑋 = 𝑈)
252unieqd 4478 . . . . . . . 8 ((𝜑𝑈 = ∅) → 𝑈 = ∅)
2624, 25eqtrd 2685 . . . . . . 7 ((𝜑𝑈 = ∅) → 𝑋 = ∅)
27 locfinref.x . . . . . . 7 𝑋 = 𝐽
2826, 27, 193eqtr3g 2708 . . . . . 6 ((𝜑𝑈 = ∅) → 𝐽 = ∅)
29 locfinref.5 . . . . . . . 8 (𝜑𝑉 ∈ (LocFin‘𝐽))
30 locfintop 21372 . . . . . . . 8 (𝑉 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
31 0top 20835 . . . . . . . 8 (𝐽 ∈ Top → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
3229, 30, 313syl 18 . . . . . . 7 (𝜑 → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
3332adantr 480 . . . . . 6 ((𝜑𝑈 = ∅) → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
3428, 33mpbid 222 . . . . 5 ((𝜑𝑈 = ∅) → 𝐽 = {∅})
3534fveq2d 6233 . . . 4 ((𝜑𝑈 = ∅) → (LocFin‘𝐽) = (LocFin‘{∅}))
3622, 35eleqtrrd 2733 . . 3 ((𝜑𝑈 = ∅) → ran ∅ ∈ (LocFin‘𝐽))
37 feq1 6064 . . . . 5 (𝑓 = ∅ → (𝑓:𝑈𝐽 ↔ ∅:𝑈𝐽))
38 rneq 5383 . . . . . 6 (𝑓 = ∅ → ran 𝑓 = ran ∅)
3938breq1d 4695 . . . . 5 (𝑓 = ∅ → (ran 𝑓Ref𝑈 ↔ ran ∅Ref𝑈))
4038eleq1d 2715 . . . . 5 (𝑓 = ∅ → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran ∅ ∈ (LocFin‘𝐽)))
4137, 39, 403anbi123d 1439 . . . 4 (𝑓 = ∅ → ((𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)) ↔ (∅:𝑈𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽))))
426, 41spcev 3331 . . 3 ((∅:𝑈𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽)) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
434, 10, 36, 42syl3anc 1366 . 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 30035 . . . 4 (𝜑 → ∃𝑔((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽))))
4847adantr 480 . . 3 ((𝜑𝑈 ≠ ∅) → ∃𝑔((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽))))
49 simpl 472 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (𝜑𝑈 ≠ ∅))
50 simprl1 1126 . . . . . . . 8 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → Fun 𝑔)
51 fdmrn 6102 . . . . . . . 8 (Fun 𝑔𝑔:dom 𝑔⟶ran 𝑔)
5250, 51sylib 208 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑔:dom 𝑔⟶ran 𝑔)
53 simprl3 1128 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔𝐽)
5452, 53fssd 6095 . . . . . 6 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑔:dom 𝑔𝐽)
55 fconstg 6130 . . . . . . . 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 20757 . . . . . . . . . 10 (𝐽 ∈ Top → ∅ ∈ 𝐽)
5829, 30, 573syl 18 . . . . . . . . 9 (𝜑 → ∅ ∈ 𝐽)
5958ad2antrr 762 . . . . . . . 8 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ∅ ∈ 𝐽)
6059snssd 4372 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → {∅} ⊆ 𝐽)
6156, 60fssd 6095 . . . . . 6 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ((𝑈 ∖ dom 𝑔) × {∅}):(𝑈 ∖ dom 𝑔)⟶𝐽)
62 disjdif 4073 . . . . . . 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 6105 . . . . . 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 1365 . . . . 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 1127 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → dom 𝑔𝑈)
67 undif 4082 . . . . . . 7 (dom 𝑔𝑈 ↔ (dom 𝑔 ∪ (𝑈 ∖ dom 𝑔)) = 𝑈)
6866, 67sylib 208 . . . . . 6 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (dom 𝑔 ∪ (𝑈 ∖ dom 𝑔)) = 𝑈)
6968feq2d 6069 . . . . 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 222 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈𝐽)
71 simpr 476 . . . . . 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 821 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔Ref𝑈)
7372adantr 480 . . . . . 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 4707 . . . . 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 476 . . . . . 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 478 . . . . . . . 8 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑈 ≠ ∅)
77 refun0 21366 . . . . . . . 8 ((ran 𝑔Ref𝑈𝑈 ≠ ∅) → (ran 𝑔 ∪ {∅})Ref𝑈)
7872, 76, 77syl2anc 694 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (ran 𝑔 ∪ {∅})Ref𝑈)
7978adantr 480 . . . . . 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 4707 . . . . 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 5601 . . . . . . 7 ran ((𝑈 ∖ dom 𝑔) × {∅}) ⊆ {∅}
82 sssn 4390 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) ⊆ {∅} ↔ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅}))
8381, 82mpbi 220 . . . . . 6 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅})
84 rnun 5576 . . . . . . . . 9 ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅}))
85 uneq2 3794 . . . . . . . . 9 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
8684, 85syl5eq 2697 . . . . . . . 8 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
87 un0 4000 . . . . . . . 8 (ran 𝑔 ∪ ∅) = ran 𝑔
8886, 87syl6eq 2701 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔)
89 uneq2 3794 . . . . . . . 8 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
9084, 89syl5eq 2697 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
9188, 90orim12i 537 . . . . . 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 844 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈)
94 simprrr 822 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔 ∈ (LocFin‘𝐽))
9594adantr 480 . . . . . 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 2730 . . . . 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 480 . . . . . . 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 8079 . . . . . . . 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 480 . . . . . . . . 9 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → ∅ ∈ 𝐽)
101100snssd 4372 . . . . . . . 8 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → {∅} ⊆ 𝐽)
102101unissd 4494 . . . . . . 7 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → {∅} ⊆ 𝐽)
103 lfinun 21376 . . . . . . 7 ((ran 𝑔 ∈ (LocFin‘𝐽) ∧ {∅} ∈ Fin ∧ {∅} ⊆ 𝐽) → (ran 𝑔 ∪ {∅}) ∈ (LocFin‘𝐽))
10497, 99, 102, 103syl3anc 1366 . . . . . 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 2730 . . . . 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 844 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))
107 refrel 21359 . . . . . . . . 9 Rel Ref
108107brrelex2i 5193 . . . . . . . 8 (𝑉Ref𝑈𝑈 ∈ V)
109 difexg 4841 . . . . . . . 8 (𝑈 ∈ V → (𝑈 ∖ dom 𝑔) ∈ V)
11046, 108, 1093syl 18 . . . . . . 7 (𝜑 → (𝑈 ∖ dom 𝑔) ∈ V)
111110adantr 480 . . . . . 6 ((𝜑𝑈 ≠ ∅) → (𝑈 ∖ dom 𝑔) ∈ V)
112 p0ex 4883 . . . . . . 7 {∅} ∈ V
113 xpexg 7002 . . . . . . 7 (((𝑈 ∖ dom 𝑔) ∈ V ∧ {∅} ∈ V) → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
114112, 113mpan2 707 . . . . . 6 ((𝑈 ∖ dom 𝑔) ∈ V → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
115 vex 3234 . . . . . . 7 𝑔 ∈ V
116 unexg 7001 . . . . . . 7 ((𝑔 ∈ V ∧ ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V)
117115, 116mpan 706 . . . . . 6 (((𝑈 ∖ dom 𝑔) × {∅}) ∈ V → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V)
118 feq1 6064 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (𝑓:𝑈𝐽 ↔ (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈𝐽))
119 rneq 5383 . . . . . . . . 9 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → ran 𝑓 = ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})))
120119breq1d 4695 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓Ref𝑈 ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈))
121119eleq1d 2715 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽)))
122118, 120, 1213anbi123d 1439 . . . . . . 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 3325 . . . . . 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 444 . . . 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 1368 . . 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 1903 . 2 ((𝜑𝑈 ≠ ∅) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
12843, 127pm2.61dane 2910 1 (𝜑 → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210   cuni 4468   class class class wbr 4685   × cxp 5141  dom cdm 5143  ran crn 5144  Fun wfun 5920  wf 5922  cfv 5926  Fincfn 7997  Topctop 20746  Refcref 21353  LocFinclocfin 21355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576  ax-ac2 9323
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-fin 8001  df-r1 8665  df-rank 8666  df-card 8803  df-ac 8977  df-top 20747  df-topon 20764  df-ref 21356  df-locfin 21358
This theorem is referenced by:  pcmplfinf  30056
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