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Theorem cmpcovf 23217
Description: Combine cmpcov 23215 with ac6sfi 9283 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 23216 . 2 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∃𝑧𝐴 𝜑)) → ∃𝑢 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑))
4 elfpw 9350 . . . 4 (𝑢 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑢𝐽𝑢 ∈ Fin))
5 simplrl 774 . . . . . . . 8 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢𝐽)
6 velpw 4599 . . . . . . . 8 (𝑢 ∈ 𝒫 𝐽𝑢𝐽)
75, 6sylibr 233 . . . . . . 7 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢 ∈ 𝒫 𝐽)
8 simplrr 775 . . . . . . 7 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢 ∈ Fin)
97, 8elind 4186 . . . . . 6 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢 ∈ (𝒫 𝐽 ∩ Fin))
10 simprl 768 . . . . . 6 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑋 = 𝑢)
11 simprr 770 . . . . . . 7 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → ∀𝑦𝑢𝑧𝐴 𝜑)
12 cmpcovf.2 . . . . . . . 8 (𝑧 = (𝑓𝑦) → (𝜑𝜓))
1312ac6sfi 9283 . . . . . . 7 ((𝑢 ∈ Fin ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))
148, 11, 13syl2anc 583 . . . . . 6 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))
15 unieq 4910 . . . . . . . . 9 (𝑠 = 𝑢 𝑠 = 𝑢)
1615eqeq2d 2735 . . . . . . . 8 (𝑠 = 𝑢 → (𝑋 = 𝑠𝑋 = 𝑢))
17 feq2 6689 . . . . . . . . . 10 (𝑠 = 𝑢 → (𝑓:𝑠𝐴𝑓:𝑢𝐴))
18 raleq 3314 . . . . . . . . . 10 (𝑠 = 𝑢 → (∀𝑦𝑠 𝜓 ↔ ∀𝑦𝑢 𝜓))
1917, 18anbi12d 630 . . . . . . . . 9 (𝑠 = 𝑢 → ((𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓) ↔ (𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓)))
2019exbidv 1916 . . . . . . . 8 (𝑠 = 𝑢 → (∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓) ↔ ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓)))
2116, 20anbi12d 630 . . . . . . 7 (𝑠 = 𝑢 → ((𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓)) ↔ (𝑋 = 𝑢 ∧ ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))))
2221rspcev 3604 . . . . . 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 412 . . . 4 ((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) → ((𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓))))
254, 24sylan2b 593 . . 3 ((𝐽 ∈ Comp ∧ 𝑢 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓))))
2625rexlimdva 3147 . 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 395   = wceq 1533  wex 1773  wcel 2098  wral 3053  wrex 3062  cin 3939  wss 3940  𝒫 cpw 4594   cuni 4899  wf 6529  cfv 6533  Fincfn 8935  Compccmp 23212
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-om 7849  df-en 8936  df-fin 8939  df-cmp 23213
This theorem is referenced by:  txtube  23466  txcmplem1  23467  txcmplem2  23468  xkococnlem  23485  cnheibor  24803  heicant  37013
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