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Theorem locfinref 30718
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 6435 . . . 4 ∅:∅⟶𝐽
2 simpr 485 . . . . 5 ((𝜑𝑈 = ∅) → 𝑈 = ∅)
32feq2d 6375 . . . 4 ((𝜑𝑈 = ∅) → (∅:𝑈𝐽 ↔ ∅:∅⟶𝐽))
41, 3mpbiri 259 . . 3 ((𝜑𝑈 = ∅) → ∅:𝑈𝐽)
5 rn0 5722 . . . . 5 ran ∅ = ∅
6 0ex 5109 . . . . . 6 ∅ ∈ V
7 refref 21809 . . . . . 6 (∅ ∈ V → ∅Ref∅)
86, 7ax-mp 5 . . . . 5 ∅Ref∅
95, 8eqbrtri 4989 . . . 4 ran ∅Ref∅
109, 2breqtrrid 5006 . . 3 ((𝜑𝑈 = ∅) → ran ∅Ref𝑈)
11 sn0top 21295 . . . . . 6 {∅} ∈ Top
1211a1i 11 . . . . 5 ((𝜑𝑈 = ∅) → {∅} ∈ Top)
13 eqidd 2798 . . . . 5 ((𝜑𝑈 = ∅) → ∅ = ∅)
14 ral0 4376 . . . . . 6 𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)
1514a1i 11 . . . . 5 ((𝜑𝑈 = ∅) → ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
166unisn 4767 . . . . . . 7 {∅} = ∅
1716eqcomi 2806 . . . . . 6 ∅ = {∅}
185unieqi 4760 . . . . . . 7 ran ∅ =
19 uni0 4778 . . . . . . 7 ∅ = ∅
2018, 19eqtr2i 2822 . . . . . 6 ∅ = ran ∅
2117, 20islocfin 21813 . . . . 5 (ran ∅ ∈ (LocFin‘{∅}) ↔ ({∅} ∈ Top ∧ ∅ = ∅ ∧ ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
2212, 13, 15, 21syl3anbrc 1336 . . . 4 ((𝜑𝑈 = ∅) → ran ∅ ∈ (LocFin‘{∅}))
23 locfinref.2 . . . . . . . . 9 (𝜑𝑋 = 𝑈)
2423adantr 481 . . . . . . . 8 ((𝜑𝑈 = ∅) → 𝑋 = 𝑈)
252unieqd 4761 . . . . . . . 8 ((𝜑𝑈 = ∅) → 𝑈 = ∅)
2624, 25eqtrd 2833 . . . . . . 7 ((𝜑𝑈 = ∅) → 𝑋 = ∅)
27 locfinref.x . . . . . . 7 𝑋 = 𝐽
2826, 27, 193eqtr3g 2856 . . . . . 6 ((𝜑𝑈 = ∅) → 𝐽 = ∅)
29 locfinref.5 . . . . . . . 8 (𝜑𝑉 ∈ (LocFin‘𝐽))
30 locfintop 21817 . . . . . . . 8 (𝑉 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
31 0top 21279 . . . . . . . 8 (𝐽 ∈ Top → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
3229, 30, 313syl 18 . . . . . . 7 (𝜑 → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
3332adantr 481 . . . . . 6 ((𝜑𝑈 = ∅) → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
3428, 33mpbid 233 . . . . 5 ((𝜑𝑈 = ∅) → 𝐽 = {∅})
3534fveq2d 6549 . . . 4 ((𝜑𝑈 = ∅) → (LocFin‘𝐽) = (LocFin‘{∅}))
3622, 35eleqtrrd 2888 . . 3 ((𝜑𝑈 = ∅) → ran ∅ ∈ (LocFin‘𝐽))
37 feq1 6370 . . . . 5 (𝑓 = ∅ → (𝑓:𝑈𝐽 ↔ ∅:𝑈𝐽))
38 rneq 5695 . . . . . 6 (𝑓 = ∅ → ran 𝑓 = ran ∅)
3938breq1d 4978 . . . . 5 (𝑓 = ∅ → (ran 𝑓Ref𝑈 ↔ ran ∅Ref𝑈))
4038eleq1d 2869 . . . . 5 (𝑓 = ∅ → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran ∅ ∈ (LocFin‘𝐽)))
4137, 39, 403anbi123d 1428 . . . 4 (𝑓 = ∅ → ((𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)) ↔ (∅:𝑈𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽))))
426, 41spcev 3551 . . 3 ((∅:𝑈𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽)) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
434, 10, 36, 42syl3anc 1364 . 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 30717 . . . 4 (𝜑 → ∃𝑔((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽))))
4847adantr 481 . . 3 ((𝜑𝑈 ≠ ∅) → ∃𝑔((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽))))
49 simpl 483 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (𝜑𝑈 ≠ ∅))
50 simprl1 1211 . . . . . . . 8 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → Fun 𝑔)
51 fdmrn 6413 . . . . . . . 8 (Fun 𝑔𝑔:dom 𝑔⟶ran 𝑔)
5250, 51sylib 219 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑔:dom 𝑔⟶ran 𝑔)
53 simprl3 1213 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔𝐽)
5452, 53fssd 6403 . . . . . 6 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑔:dom 𝑔𝐽)
55 fconstg 6441 . . . . . . . 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 21200 . . . . . . . . . 10 (𝐽 ∈ Top → ∅ ∈ 𝐽)
5829, 30, 573syl 18 . . . . . . . . 9 (𝜑 → ∅ ∈ 𝐽)
5958ad2antrr 722 . . . . . . . 8 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ∅ ∈ 𝐽)
6059snssd 4655 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → {∅} ⊆ 𝐽)
6156, 60fssd 6403 . . . . . 6 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ((𝑈 ∖ dom 𝑔) × {∅}):(𝑈 ∖ dom 𝑔)⟶𝐽)
62 disjdif 4341 . . . . . . 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 6416 . . . . . 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 834 . . . . 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 1212 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → dom 𝑔𝑈)
67 undif 4350 . . . . . . 7 (dom 𝑔𝑈 ↔ (dom 𝑔 ∪ (𝑈 ∖ dom 𝑔)) = 𝑈)
6866, 67sylib 219 . . . . . 6 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (dom 𝑔 ∪ (𝑈 ∖ dom 𝑔)) = 𝑈)
6968feq2d 6375 . . . . 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 233 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈𝐽)
71 simpr 485 . . . . . 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 777 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔Ref𝑈)
7372adantr 481 . . . . . 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 4990 . . . . 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 485 . . . . . 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 496 . . . . . . . 8 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑈 ≠ ∅)
77 refun0 21811 . . . . . . . 8 ((ran 𝑔Ref𝑈𝑈 ≠ ∅) → (ran 𝑔 ∪ {∅})Ref𝑈)
7872, 76, 77syl2anc 584 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (ran 𝑔 ∪ {∅})Ref𝑈)
7978adantr 481 . . . . . 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 4990 . . . . 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 5912 . . . . . . 7 ran ((𝑈 ∖ dom 𝑔) × {∅}) ⊆ {∅}
82 sssn 4672 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) ⊆ {∅} ↔ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅}))
8381, 82mpbi 231 . . . . . 6 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅})
84 rnun 5887 . . . . . . . . 9 ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅}))
85 uneq2 4060 . . . . . . . . 9 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
8684, 85syl5eq 2845 . . . . . . . 8 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
87 un0 4270 . . . . . . . 8 (ran 𝑔 ∪ ∅) = ran 𝑔
8886, 87syl6eq 2849 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔)
89 uneq2 4060 . . . . . . . 8 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
9084, 89syl5eq 2845 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
9188, 90orim12i 903 . . . . . 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 953 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈)
94 simprrr 778 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔 ∈ (LocFin‘𝐽))
9594adantr 481 . . . . . 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 2885 . . . . 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 481 . . . . . . 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 8449 . . . . . . . 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 481 . . . . . . . . 9 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → ∅ ∈ 𝐽)
101100snssd 4655 . . . . . . . 8 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → {∅} ⊆ 𝐽)
102101unissd 4775 . . . . . . 7 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → {∅} ⊆ 𝐽)
103 lfinun 21821 . . . . . . 7 ((ran 𝑔 ∈ (LocFin‘𝐽) ∧ {∅} ∈ Fin ∧ {∅} ⊆ 𝐽) → (ran 𝑔 ∪ {∅}) ∈ (LocFin‘𝐽))
10497, 99, 102, 103syl3anc 1364 . . . . . 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 2885 . . . . 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 953 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))
107 refrel 21804 . . . . . . . . 9 Rel Ref
108107brrelex2i 5502 . . . . . . . 8 (𝑉Ref𝑈𝑈 ∈ V)
109 difexg 5129 . . . . . . . 8 (𝑈 ∈ V → (𝑈 ∖ dom 𝑔) ∈ V)
11046, 108, 1093syl 18 . . . . . . 7 (𝜑 → (𝑈 ∖ dom 𝑔) ∈ V)
111110adantr 481 . . . . . 6 ((𝜑𝑈 ≠ ∅) → (𝑈 ∖ dom 𝑔) ∈ V)
112 p0ex 5182 . . . . . . 7 {∅} ∈ V
113 xpexg 7337 . . . . . . 7 (((𝑈 ∖ dom 𝑔) ∈ V ∧ {∅} ∈ V) → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
114112, 113mpan2 687 . . . . . 6 ((𝑈 ∖ dom 𝑔) ∈ V → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
115 vex 3443 . . . . . . 7 𝑔 ∈ V
116 unexg 7336 . . . . . . 7 ((𝑔 ∈ V ∧ ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V)
117115, 116mpan 686 . . . . . 6 (((𝑈 ∖ dom 𝑔) × {∅}) ∈ V → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V)
118 feq1 6370 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (𝑓:𝑈𝐽 ↔ (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈𝐽))
119 rneq 5695 . . . . . . . . 9 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → ran 𝑓 = ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})))
120119breq1d 4978 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓Ref𝑈 ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈))
121119eleq1d 2869 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽)))
122118, 120, 1213anbi123d 1428 . . . . . . 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 3542 . . . . . 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 407 . . . 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 1365 . . 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 1917 . 2 ((𝜑𝑈 ≠ ∅) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
12843, 127pm2.61dane 3074 1 (𝜑 → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 842  w3a 1080   = wceq 1525  wex 1765  wcel 2083  wne 2986  wral 3107  wrex 3108  {crab 3111  Vcvv 3440  cdif 3862  cun 3863  cin 3864  wss 3865  c0 4217  {csn 4478   cuni 4751   class class class wbr 4968   × cxp 5448  dom cdm 5450  ran crn 5451  Fun wfun 6226  wf 6228  cfv 6232  Fincfn 8364  Topctop 21189  Refcref 21798  LocFinclocfin 21800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-reg 8909  ax-inf2 8957  ax-ac2 9738
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-iin 4834  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-oadd 7964  df-er 8146  df-en 8365  df-dom 8366  df-fin 8368  df-r1 9046  df-rank 9047  df-card 9221  df-ac 9395  df-top 21190  df-topon 21207  df-ref 21801  df-locfin 21803
This theorem is referenced by:  pcmplfinf  30738
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