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Theorem locfinref 33801
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 6789 . . . 4 ∅:∅⟶𝐽
2 simpr 484 . . . . 5 ((𝜑𝑈 = ∅) → 𝑈 = ∅)
32feq2d 6722 . . . 4 ((𝜑𝑈 = ∅) → (∅:𝑈𝐽 ↔ ∅:∅⟶𝐽))
41, 3mpbiri 258 . . 3 ((𝜑𝑈 = ∅) → ∅:𝑈𝐽)
5 rn0 5938 . . . . 5 ran ∅ = ∅
6 0ex 5312 . . . . . 6 ∅ ∈ V
7 refref 23536 . . . . . 6 (∅ ∈ V → ∅Ref∅)
86, 7ax-mp 5 . . . . 5 ∅Ref∅
95, 8eqbrtri 5168 . . . 4 ran ∅Ref∅
109, 2breqtrrid 5185 . . 3 ((𝜑𝑈 = ∅) → ran ∅Ref𝑈)
11 sn0top 23021 . . . . . 6 {∅} ∈ Top
1211a1i 11 . . . . 5 ((𝜑𝑈 = ∅) → {∅} ∈ Top)
13 eqidd 2735 . . . . 5 ((𝜑𝑈 = ∅) → ∅ = ∅)
14 ral0 4518 . . . . . 6 𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)
1514a1i 11 . . . . 5 ((𝜑𝑈 = ∅) → ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
166unisn 4930 . . . . . . 7 {∅} = ∅
1716eqcomi 2743 . . . . . 6 ∅ = {∅}
185unieqi 4923 . . . . . . 7 ran ∅ =
19 uni0 4939 . . . . . . 7 ∅ = ∅
2018, 19eqtr2i 2763 . . . . . 6 ∅ = ran ∅
2117, 20islocfin 23540 . . . . 5 (ran ∅ ∈ (LocFin‘{∅}) ↔ ({∅} ∈ Top ∧ ∅ = ∅ ∧ ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
2212, 13, 15, 21syl3anbrc 1342 . . . 4 ((𝜑𝑈 = ∅) → ran ∅ ∈ (LocFin‘{∅}))
23 locfinref.2 . . . . . . . . 9 (𝜑𝑋 = 𝑈)
2423adantr 480 . . . . . . . 8 ((𝜑𝑈 = ∅) → 𝑋 = 𝑈)
252unieqd 4924 . . . . . . . 8 ((𝜑𝑈 = ∅) → 𝑈 = ∅)
2624, 25eqtrd 2774 . . . . . . 7 ((𝜑𝑈 = ∅) → 𝑋 = ∅)
27 locfinref.x . . . . . . 7 𝑋 = 𝐽
2826, 27, 193eqtr3g 2797 . . . . . 6 ((𝜑𝑈 = ∅) → 𝐽 = ∅)
29 locfinref.5 . . . . . . . 8 (𝜑𝑉 ∈ (LocFin‘𝐽))
30 locfintop 23544 . . . . . . . 8 (𝑉 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
31 0top 23005 . . . . . . . 8 (𝐽 ∈ Top → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
3229, 30, 313syl 18 . . . . . . 7 (𝜑 → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
3332adantr 480 . . . . . 6 ((𝜑𝑈 = ∅) → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
3428, 33mpbid 232 . . . . 5 ((𝜑𝑈 = ∅) → 𝐽 = {∅})
3534fveq2d 6910 . . . 4 ((𝜑𝑈 = ∅) → (LocFin‘𝐽) = (LocFin‘{∅}))
3622, 35eleqtrrd 2841 . . 3 ((𝜑𝑈 = ∅) → ran ∅ ∈ (LocFin‘𝐽))
37 feq1 6716 . . . . 5 (𝑓 = ∅ → (𝑓:𝑈𝐽 ↔ ∅:𝑈𝐽))
38 rneq 5949 . . . . . 6 (𝑓 = ∅ → ran 𝑓 = ran ∅)
3938breq1d 5157 . . . . 5 (𝑓 = ∅ → (ran 𝑓Ref𝑈 ↔ ran ∅Ref𝑈))
4038eleq1d 2823 . . . . 5 (𝑓 = ∅ → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran ∅ ∈ (LocFin‘𝐽)))
4137, 39, 403anbi123d 1435 . . . 4 (𝑓 = ∅ → ((𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)) ↔ (∅:𝑈𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽))))
426, 41spcev 3605 . . 3 ((∅:𝑈𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽)) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
434, 10, 36, 42syl3anc 1370 . 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 33800 . . . 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 482 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (𝜑𝑈 ≠ ∅))
50 simprl1 1217 . . . . . . . 8 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → Fun 𝑔)
51 fdmrn 6767 . . . . . . . 8 (Fun 𝑔𝑔:dom 𝑔⟶ran 𝑔)
5250, 51sylib 218 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑔:dom 𝑔⟶ran 𝑔)
53 simprl3 1219 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran 𝑔𝐽)
5452, 53fssd 6753 . . . . . 6 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑔:dom 𝑔𝐽)
55 fconstg 6795 . . . . . . . 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 22925 . . . . . . . . . 10 (𝐽 ∈ Top → ∅ ∈ 𝐽)
5829, 30, 573syl 18 . . . . . . . . 9 (𝜑 → ∅ ∈ 𝐽)
5958ad2antrr 726 . . . . . . . 8 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ∅ ∈ 𝐽)
6059snssd 4813 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → {∅} ⊆ 𝐽)
6156, 60fssd 6753 . . . . . 6 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ((𝑈 ∖ dom 𝑔) × {∅}):(𝑈 ∖ dom 𝑔)⟶𝐽)
62 disjdif 4477 . . . . . . 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 6771 . . . . . 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 838 . . . . 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 1218 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → dom 𝑔𝑈)
67 undif 4487 . . . . . . 7 (dom 𝑔𝑈 ↔ (dom 𝑔 ∪ (𝑈 ∖ dom 𝑔)) = 𝑈)
6866, 67sylib 218 . . . . . 6 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (dom 𝑔 ∪ (𝑈 ∖ dom 𝑔)) = 𝑈)
6968feq2d 6722 . . . . 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 232 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈𝐽)
71 simpr 484 . . . . . 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 781 . . . . . . 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 5169 . . . . 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 484 . . . . . 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 495 . . . . . . . 8 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → 𝑈 ≠ ∅)
77 refun0 23538 . . . . . . . 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 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 5169 . . . . 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 6193 . . . . . . 7 ran ((𝑈 ∖ dom 𝑔) × {∅}) ⊆ {∅}
82 sssn 4830 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) ⊆ {∅} ↔ (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅}))
8381, 82mpbi 230 . . . . . 6 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ ∨ ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅})
84 rnun 6167 . . . . . . . . 9 ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅}))
85 uneq2 4171 . . . . . . . . 9 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
8684, 85eqtrid 2786 . . . . . . . 8 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
87 un0 4399 . . . . . . . 8 (ran 𝑔 ∪ ∅) = ran 𝑔
8886, 87eqtrdi 2790 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔)
89 uneq2 4171 . . . . . . . 8 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
9084, 89eqtrid 2786 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
9188, 90orim12i 908 . . . . . 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 960 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈)
94 simprrr 782 . . . . . . 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 2838 . . . . 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 9081 . . . . . . . 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 4813 . . . . . . . 8 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → {∅} ⊆ 𝐽)
102101unissd 4921 . . . . . . 7 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → {∅} ⊆ 𝐽)
103 lfinun 23548 . . . . . . 7 ((ran 𝑔 ∈ (LocFin‘𝐽) ∧ {∅} ∈ Fin ∧ {∅} ⊆ 𝐽) → (ran 𝑔 ∪ {∅}) ∈ (LocFin‘𝐽))
10497, 99, 102, 103syl3anc 1370 . . . . . 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 2838 . . . . 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 960 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))
107 refrel 23531 . . . . . . . . 9 Rel Ref
108107brrelex2i 5745 . . . . . . . 8 (𝑉Ref𝑈𝑈 ∈ V)
109 difexg 5334 . . . . . . . 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 5389 . . . . . . 7 {∅} ∈ V
113 xpexg 7768 . . . . . . 7 (((𝑈 ∖ dom 𝑔) ∈ V ∧ {∅} ∈ V) → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
114112, 113mpan2 691 . . . . . 6 ((𝑈 ∖ dom 𝑔) ∈ V → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
115 vex 3481 . . . . . . 7 𝑔 ∈ V
116 unexg 7761 . . . . . . 7 ((𝑔 ∈ V ∧ ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V) → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V)
117115, 116mpan 690 . . . . . 6 (((𝑈 ∖ dom 𝑔) × {∅}) ∈ V → (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ V)
118 feq1 6716 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (𝑓:𝑈𝐽 ↔ (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})):𝑈𝐽))
119 rneq 5949 . . . . . . . . 9 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → ran 𝑓 = ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})))
120119breq1d 5157 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓Ref𝑈 ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈))
121119eleq1d 2823 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽)))
122118, 120, 1213anbi123d 1435 . . . . . . 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 3596 . . . . . 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 406 . . . 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 1371 . . 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 1932 . 2 ((𝜑𝑈 ≠ ∅) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
12843, 127pm2.61dane 3026 1 (𝜑 → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  cdif 3959  cun 3960  cin 3961  wss 3962  c0 4338  {csn 4630   cuni 4911   class class class wbr 5147   × cxp 5686  dom cdm 5688  ran crn 5689  Fun wfun 6556  wf 6558  cfv 6562  Fincfn 8983  Topctop 22914  Refcref 23525  LocFinclocfin 23527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-reg 9629  ax-inf2 9678  ax-ac2 10500
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-fin 8987  df-r1 9801  df-rank 9802  df-card 9976  df-ac 10153  df-top 22915  df-topon 22932  df-ref 23528  df-locfin 23530
This theorem is referenced by:  pcmplfinf  33821
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