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Theorem cmpcovf 21004
 Description: Combine cmpcov 21002 with ac6sfi 8089 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 472 . 2 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∃𝑧𝐴 𝜑)) → 𝐽 ∈ Comp)
2 iscmp.1 . . 3 𝑋 = 𝐽
32cmpcov2 21003 . 2 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦 ∧ ∃𝑧𝐴 𝜑)) → ∃𝑢 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑))
4 elfpw 8151 . . . 4 (𝑢 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑢𝐽𝑢 ∈ Fin))
5 simplrl 796 . . . . . . . 8 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢𝐽)
6 selpw 4115 . . . . . . . 8 (𝑢 ∈ 𝒫 𝐽𝑢𝐽)
75, 6sylibr 223 . . . . . . 7 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢 ∈ 𝒫 𝐽)
8 simplrr 797 . . . . . . 7 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢 ∈ Fin)
97, 8elind 3760 . . . . . 6 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑢 ∈ (𝒫 𝐽 ∩ Fin))
10 simprl 790 . . . . . 6 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → 𝑋 = 𝑢)
11 simprr 792 . . . . . . 7 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → ∀𝑦𝑢𝑧𝐴 𝜑)
12 cmpcovf.2 . . . . . . . 8 (𝑧 = (𝑓𝑦) → (𝜑𝜓))
1312ac6sfi 8089 . . . . . . 7 ((𝑢 ∈ Fin ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))
148, 11, 13syl2anc 691 . . . . . 6 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))
15 unieq 4380 . . . . . . . . 9 (𝑠 = 𝑢 𝑠 = 𝑢)
1615eqeq2d 2620 . . . . . . . 8 (𝑠 = 𝑢 → (𝑋 = 𝑠𝑋 = 𝑢))
17 feq2 5940 . . . . . . . . . 10 (𝑠 = 𝑢 → (𝑓:𝑠𝐴𝑓:𝑢𝐴))
18 raleq 3115 . . . . . . . . . 10 (𝑠 = 𝑢 → (∀𝑦𝑠 𝜓 ↔ ∀𝑦𝑢 𝜓))
1917, 18anbi12d 743 . . . . . . . . 9 (𝑠 = 𝑢 → ((𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓) ↔ (𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓)))
2019exbidv 1837 . . . . . . . 8 (𝑠 = 𝑢 → (∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓) ↔ ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓)))
2116, 20anbi12d 743 . . . . . . 7 (𝑠 = 𝑢 → ((𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓)) ↔ (𝑋 = 𝑢 ∧ ∃𝑓(𝑓:𝑢𝐴 ∧ ∀𝑦𝑢 𝜓))))
2221rspcev 3282 . . . . . 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 1316 . . . . 5 (((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) ∧ (𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓)))
2423ex 449 . . . 4 ((𝐽 ∈ Comp ∧ (𝑢𝐽𝑢 ∈ Fin)) → ((𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓))))
254, 24sylan2b 491 . . 3 ((𝐽 ∈ Comp ∧ 𝑢 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = 𝑢 ∧ ∀𝑦𝑢𝑧𝐴 𝜑) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∃𝑓(𝑓:𝑠𝐴 ∧ ∀𝑦𝑠 𝜓))))
2625rexlimdva 3013 . 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 63 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 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ⟶wf 5800  ‘cfv 5804  Fincfn 7841  Compccmp 20999 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-er 7629  df-en 7842  df-fin 7845  df-cmp 21000 This theorem is referenced by:  txtube  21253  txcmplem1  21254  txcmplem2  21255  xkococnlem  21272  cnheibor  22562  heicant  32614
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