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Theorem cmpcovf 23399
Description: Combine cmpcov 23397 with ac6sfi 9320 to show the existence of a function that indexes the elements that are generating the open cover. (Contributed by Mario Carneiro, 14-Sep-2014.)
Hypotheses
Ref Expression
iscmp.1 𝑋 = 𝐽
cmpcovf.2 (𝑧 = (𝑓𝑦) → (𝜑𝜓))
Assertion
Ref Expression
cmpcovf ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∃𝑧𝐴 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓)))
Distinct variable groups:   𝑓,𝑠,𝑥,𝑦,𝑧,𝐴   𝐽,𝑠,𝑥,𝑦,𝑧   𝜑,𝑓,𝑠,𝑥   𝜓,𝑠,𝑧   𝑥,𝑋,𝑠
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝜓(𝑥,𝑦,𝑓)   𝐽(𝑓)   𝑋(𝑦,𝑧,𝑓)

Proof of Theorem cmpcovf
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 simpl 482 . 2 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∃𝑧𝐴 𝜑)) → 𝐽 ∈ Comp)
2 iscmp.1 . . 3 𝑋 = 𝐽
32cmpcov2 23398 . 2 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∃𝑧𝐴 𝜑)) → ∃𝑢 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑))
4 elfpw 9394 . . . 4 (𝑢 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑢𝐽𝑢 ∈ Fin))
5 simplrl 777 . . . . . . . 8 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢𝐽)
6 velpw 4605 . . . . . . . 8 (𝑢 ∈ 𝒫 𝐽𝑢𝐽)
75, 6sylibr 234 . . . . . . 7 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢 ∈ 𝒫 𝐽)
8 simplrr 778 . . . . . . 7 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢 ∈ Fin)
97, 8elind 4200 . . . . . 6 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢 ∈ (𝒫 𝐽 ∩ Fin))
10 simprl 771 . . . . . 6 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑋 = 𝑢)
11 simprr 773 . . . . . . 7 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → ∀𝑦𝑢𝑧𝐴 𝜑)
12 cmpcovf.2 . . . . . . . 8 (𝑧 = (𝑓𝑦) → (𝜑𝜓))
1312ac6sfi 9320 . . . . . . 7 ((𝑢 ∈ Fin ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))
148, 11, 13syl2anc 584 . . . . . 6 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))
15 unieq 4918 . . . . . . . . 9 (𝑠 = 𝑢 𝑠 = 𝑢)
1615eqeq2d 2748 . . . . . . . 8 (𝑠 = 𝑢 → (𝑋 = 𝑠𝑋 = 𝑢))
17 feq2 6717 . . . . . . . . . 10 (𝑠 = 𝑢 → (𝑓:𝑠𝐴𝑓:𝑢𝐴))
18 raleq 3323 . . . . . . . . . 10 (𝑠 = 𝑢 → (∀𝑦𝑠 𝜓 ↔ ∀𝑦𝑢 𝜓))
1917, 18anbi12d 632 . . . . . . . . 9 (𝑠 = 𝑢 → ((𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓) ↔ (𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓)))
2019exbidv 1921 . . . . . . . 8 (𝑠 = 𝑢 → (∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓) ↔ ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓)))
2116, 20anbi12d 632 . . . . . . 7 (𝑠 = 𝑢 → ((𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓)) ↔ (𝑋 = 𝑢 ∧ ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))))
2221rspcev 3622 . . . . . 6 ((𝑢 ∈ (𝒫 𝐽 ∩ Fin) ∧ (𝑋 = 𝑢 ∧ ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓)))
239, 10, 14, 22syl12anc 837 . . . . 5 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓)))
2423ex 412 . . . 4 ((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) → ((𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓))))
254, 24sylan2b 594 . . 3 ((𝐽 ∈ Comp ∧ 𝑢 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓))))
2625rexlimdva 3155 . 2 (𝐽 ∈ Comp → (∃𝑢 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓))))
271, 3, 26sylc 65 1 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∃𝑧𝐴 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907  wf 6557  cfv 6561  Fincfn 8985  Compccmp 23394
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-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
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-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  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-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  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-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-om 7888  df-en 8986  df-fin 8989  df-cmp 23395
This theorem is referenced by:  txtube  23648  txcmplem1  23649  txcmplem2  23650  xkococnlem  23667  cnheibor  24987  heicant  37662
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