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Theorem cmpcovf 21991
Description: Combine cmpcov 21989 with ac6sfi 8754 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 485 . 2 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∃𝑧𝐴 𝜑)) → 𝐽 ∈ Comp)
2 iscmp.1 . . 3 𝑋 = 𝐽
32cmpcov2 21990 . 2 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∃𝑧𝐴 𝜑)) → ∃𝑢 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑))
4 elfpw 8818 . . . 4 (𝑢 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑢𝐽𝑢 ∈ Fin))
5 simplrl 775 . . . . . . . 8 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢𝐽)
6 velpw 4545 . . . . . . . 8 (𝑢 ∈ 𝒫 𝐽𝑢𝐽)
75, 6sylibr 236 . . . . . . 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 4169 . . . . . 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 8754 . . . . . . 7 ((𝑢 ∈ Fin ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))
148, 11, 13syl2anc 586 . . . . . 6 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))
15 unieq 4838 . . . . . . . . 9 (𝑠 = 𝑢 𝑠 = 𝑢)
1615eqeq2d 2830 . . . . . . . 8 (𝑠 = 𝑢 → (𝑋 = 𝑠𝑋 = 𝑢))
17 feq2 6489 . . . . . . . . . 10 (𝑠 = 𝑢 → (𝑓:𝑠𝐴𝑓:𝑢𝐴))
18 raleq 3404 . . . . . . . . . 10 (𝑠 = 𝑢 → (∀𝑦𝑠 𝜓 ↔ ∀𝑦𝑢 𝜓))
1917, 18anbi12d 632 . . . . . . . . 9 (𝑠 = 𝑢 → ((𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓) ↔ (𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓)))
2019exbidv 1915 . . . . . . . 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 3621 . . . . . 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 834 . . . . 5 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓)))
2423ex 415 . . . 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 3282 . 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 208  wa 398   = wceq 1530  wex 1773  wcel 2107  wral 3136  wrex 3137  cin 3933  wss 3934  𝒫 cpw 4537   cuni 4830  wf 6344  cfv 6348  Fincfn 8501  Compccmp 21986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7573  df-1o 8094  df-er 8281  df-en 8502  df-fin 8505  df-cmp 21987
This theorem is referenced by:  txtube  22240  txcmplem1  22241  txcmplem2  22242  xkococnlem  22259  cnheibor  23551  heicant  34914
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