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Theorem cmpcovf 22587
Description: Combine cmpcov 22585 with ac6sfi 9102 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 484 . 2 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∃𝑧𝐴 𝜑)) → 𝐽 ∈ Comp)
2 iscmp.1 . . 3 𝑋 = 𝐽
32cmpcov2 22586 . 2 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∃𝑧𝐴 𝜑)) → ∃𝑢 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑))
4 elfpw 9165 . . . 4 (𝑢 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑢𝐽𝑢 ∈ Fin))
5 simplrl 775 . . . . . . . 8 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢𝐽)
6 velpw 4544 . . . . . . . 8 (𝑢 ∈ 𝒫 𝐽𝑢𝐽)
75, 6sylibr 233 . . . . . . 7 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢 ∈ 𝒫 𝐽)
8 simplrr 776 . . . . . . 7 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢 ∈ Fin)
97, 8elind 4134 . . . . . 6 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢 ∈ (𝒫 𝐽 ∩ Fin))
10 simprl 769 . . . . . 6 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑋 = 𝑢)
11 simprr 771 . . . . . . 7 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → ∀𝑦𝑢𝑧𝐴 𝜑)
12 cmpcovf.2 . . . . . . . 8 (𝑧 = (𝑓𝑦) → (𝜑𝜓))
1312ac6sfi 9102 . . . . . . 7 ((𝑢 ∈ Fin ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))
148, 11, 13syl2anc 585 . . . . . 6 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))
15 unieq 4855 . . . . . . . . 9 (𝑠 = 𝑢 𝑠 = 𝑢)
1615eqeq2d 2747 . . . . . . . 8 (𝑠 = 𝑢 → (𝑋 = 𝑠𝑋 = 𝑢))
17 feq2 6612 . . . . . . . . . 10 (𝑠 = 𝑢 → (𝑓:𝑠𝐴𝑓:𝑢𝐴))
18 raleq 3354 . . . . . . . . . 10 (𝑠 = 𝑢 → (∀𝑦𝑠 𝜓 ↔ ∀𝑦𝑢 𝜓))
1917, 18anbi12d 632 . . . . . . . . 9 (𝑠 = 𝑢 → ((𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓) ↔ (𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓)))
2019exbidv 1922 . . . . . . . 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 3566 . . . . . 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 835 . . . . 5 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓)))
2423ex 414 . . . 4 ((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) → ((𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓))))
254, 24sylan2b 595 . . 3 ((𝐽 ∈ Comp ∧ 𝑢 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓))))
2625rexlimdva 3149 . 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 205  wa 397   = wceq 1539  wex 1779  wcel 2104  wral 3062  wrex 3071  cin 3891  wss 3892  𝒫 cpw 4539   cuni 4844  wf 6454  cfv 6458  Fincfn 8764  Compccmp 22582
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-om 7745  df-en 8765  df-fin 8768  df-cmp 22583
This theorem is referenced by:  txtube  22836  txcmplem1  22837  txcmplem2  22838  xkococnlem  22855  cnheibor  24163  heicant  35856
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