Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  locfinref Structured version   Visualization version   GIF version

Theorem locfinref 33840
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 5936 . . . . 5 ran ∅ = ∅
6 0ex 5307 . . . . . 6 ∅ ∈ V
7 refref 23521 . . . . . 6 (∅ ∈ V → ∅Ref∅)
86, 7ax-mp 5 . . . . 5 ∅Ref∅
95, 8eqbrtri 5164 . . . 4 ran ∅Ref∅
109, 2breqtrrid 5181 . . 3 ((𝜑𝑈 = ∅) → ran ∅Ref𝑈)
11 sn0top 23006 . . . . . 6 {∅} ∈ Top
1211a1i 11 . . . . 5 ((𝜑𝑈 = ∅) → {∅} ∈ Top)
13 eqidd 2738 . . . . 5 ((𝜑𝑈 = ∅) → ∅ = ∅)
14 ral0 4513 . . . . . 6 𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)
1514a1i 11 . . . . 5 ((𝜑𝑈 = ∅) → ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
166unisn 4926 . . . . . . 7 {∅} = ∅
1716eqcomi 2746 . . . . . 6 ∅ = {∅}
185unieqi 4919 . . . . . . 7 ran ∅ =
19 uni0 4935 . . . . . . 7 ∅ = ∅
2018, 19eqtr2i 2766 . . . . . 6 ∅ = ran ∅
2117, 20islocfin 23525 . . . . 5 (ran ∅ ∈ (LocFin‘{∅}) ↔ ({∅} ∈ Top ∧ ∅ = ∅ ∧ ∀𝑥 ∈ ∅ ∃𝑛 ∈ {∅} (𝑥𝑛 ∧ {𝑠 ∈ ran ∅ ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
2212, 13, 15, 21syl3anbrc 1344 . . . 4 ((𝜑𝑈 = ∅) → ran ∅ ∈ (LocFin‘{∅}))
23 locfinref.2 . . . . . . . . 9 (𝜑𝑋 = 𝑈)
2423adantr 480 . . . . . . . 8 ((𝜑𝑈 = ∅) → 𝑋 = 𝑈)
252unieqd 4920 . . . . . . . 8 ((𝜑𝑈 = ∅) → 𝑈 = ∅)
2624, 25eqtrd 2777 . . . . . . 7 ((𝜑𝑈 = ∅) → 𝑋 = ∅)
27 locfinref.x . . . . . . 7 𝑋 = 𝐽
2826, 27, 193eqtr3g 2800 . . . . . 6 ((𝜑𝑈 = ∅) → 𝐽 = ∅)
29 locfinref.5 . . . . . . . 8 (𝜑𝑉 ∈ (LocFin‘𝐽))
30 locfintop 23529 . . . . . . . 8 (𝑉 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
31 0top 22990 . . . . . . . 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 2844 . . 3 ((𝜑𝑈 = ∅) → ran ∅ ∈ (LocFin‘𝐽))
37 feq1 6716 . . . . 5 (𝑓 = ∅ → (𝑓:𝑈𝐽 ↔ ∅:𝑈𝐽))
38 rneq 5947 . . . . . 6 (𝑓 = ∅ → ran 𝑓 = ran ∅)
3938breq1d 5153 . . . . 5 (𝑓 = ∅ → (ran 𝑓Ref𝑈 ↔ ran ∅Ref𝑈))
4038eleq1d 2826 . . . . 5 (𝑓 = ∅ → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran ∅ ∈ (LocFin‘𝐽)))
4137, 39, 403anbi123d 1438 . . . 4 (𝑓 = ∅ → ((𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)) ↔ (∅:𝑈𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽))))
426, 41spcev 3606 . . 3 ((∅:𝑈𝐽 ∧ ran ∅Ref𝑈 ∧ ran ∅ ∈ (LocFin‘𝐽)) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
434, 10, 36, 42syl3anc 1373 . 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 33839 . . . 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 1219 . . . . . . . 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 1221 . . . . . . 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 22910 . . . . . . . . . 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 4809 . . . . . . 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 4472 . . . . . . 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 1220 . . . . . . 7 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → dom 𝑔𝑈)
67 undif 4482 . . . . . . 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 5165 . . . . 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 23523 . . . . . . . 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 5165 . . . . 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 6192 . . . . . . 7 ran ((𝑈 ∖ dom 𝑔) × {∅}) ⊆ {∅}
82 sssn 4826 . . . . . . 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 6165 . . . . . . . . 9 ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅}))
85 uneq2 4162 . . . . . . . . 9 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
8684, 85eqtrid 2789 . . . . . . . 8 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ ∅))
87 un0 4394 . . . . . . . 8 (ran 𝑔 ∪ ∅) = ran 𝑔
8886, 87eqtrdi 2793 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = ∅ → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = ran 𝑔)
89 uneq2 4162 . . . . . . . 8 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → (ran 𝑔 ∪ ran ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
9084, 89eqtrid 2789 . . . . . . 7 (ran ((𝑈 ∖ dom 𝑔) × {∅}) = {∅} → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅}))
9188, 90orim12i 909 . . . . . 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 961 . . . 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 2841 . . . . 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 9083 . . . . . . . 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 4809 . . . . . . . 8 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → {∅} ⊆ 𝐽)
102101unissd 4917 . . . . . . 7 ((((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) ∧ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) = (ran 𝑔 ∪ {∅})) → {∅} ⊆ 𝐽)
103 lfinun 23533 . . . . . . 7 ((ran 𝑔 ∈ (LocFin‘𝐽) ∧ {∅} ∈ Fin ∧ {∅} ⊆ 𝐽) → (ran 𝑔 ∪ {∅}) ∈ (LocFin‘𝐽))
10497, 99, 102, 103syl3anc 1373 . . . . . 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 2841 . . . . 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 961 . . . 4 (((𝜑𝑈 ≠ ∅) ∧ ((Fun 𝑔 ∧ dom 𝑔𝑈 ∧ ran 𝑔𝐽) ∧ (ran 𝑔Ref𝑈 ∧ ran 𝑔 ∈ (LocFin‘𝐽)))) → ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽))
107 refrel 23516 . . . . . . . . 9 Rel Ref
108107brrelex2i 5742 . . . . . . . 8 (𝑉Ref𝑈𝑈 ∈ V)
109 difexg 5329 . . . . . . . 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 5384 . . . . . . 7 {∅} ∈ V
113 xpexg 7770 . . . . . . 7 (((𝑈 ∖ dom 𝑔) ∈ V ∧ {∅} ∈ V) → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
114112, 113mpan2 691 . . . . . 6 ((𝑈 ∖ dom 𝑔) ∈ V → ((𝑈 ∖ dom 𝑔) × {∅}) ∈ V)
115 vex 3484 . . . . . . 7 𝑔 ∈ V
116 unexg 7763 . . . . . . 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 5947 . . . . . . . . 9 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → ran 𝑓 = ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})))
120119breq1d 5153 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓Ref𝑈 ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅}))Ref𝑈))
121119eleq1d 2826 . . . . . . . 8 (𝑓 = (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) → (ran 𝑓 ∈ (LocFin‘𝐽) ↔ ran (𝑔 ∪ ((𝑈 ∖ dom 𝑔) × {∅})) ∈ (LocFin‘𝐽)))
122118, 120, 1213anbi123d 1438 . . . . . . 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 3597 . . . . . 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 1374 . . 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 1935 . 2 ((𝜑𝑈 ≠ ∅) → ∃𝑓(𝑓:𝑈𝐽 ∧ ran 𝑓Ref𝑈 ∧ ran 𝑓 ∈ (LocFin‘𝐽)))
12843, 127pm2.61dane 3029 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 848  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626   cuni 4907   class class class wbr 5143   × cxp 5683  dom cdm 5685  ran crn 5686  Fun wfun 6555  wf 6557  cfv 6561  Fincfn 8985  Topctop 22899  Refcref 23510  LocFinclocfin 23512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-fin 8989  df-r1 9804  df-rank 9805  df-card 9979  df-ac 10156  df-top 22900  df-topon 22917  df-ref 23513  df-locfin 23515
This theorem is referenced by:  pcmplfinf  33860
  Copyright terms: Public domain W3C validator