Theorem cmpcovf 22895

Description: Combine cmpcov 22893 with ac6sfi 9287 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 = (𝑓‘𝑦) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cmpcovf	⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝐴 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 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.

Step	Hyp	Ref	Expression
1		simpl 484	. 2 ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝐴 𝜑)) → 𝐽 ∈ Comp)
2		iscmp.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
3	2	cmpcov2 22894	. 2 ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝐴 𝜑)) → ∃𝑢 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑))
4		elfpw 9354	. . . 4 ⊢ (𝑢 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin))
5		simplrl 776	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ⊆ 𝐽)
6		velpw 4608	. . . . . . . 8 ⊢ (𝑢 ∈ 𝒫 𝐽 ↔ 𝑢 ⊆ 𝐽)
7	5, 6	sylibr 233	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ 𝒫 𝐽)
8		simplrr 777	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ Fin)
9	7, 8	elind 4195	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ (𝒫 𝐽 ∩ Fin))
10		simprl 770	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑋 = ∪ 𝑢)
11		simprr 772	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)
12		cmpcovf.2	. . . . . . . 8 ⊢ (𝑧 = (𝑓‘𝑦) → (𝜑 ↔ 𝜓))
13	12	ac6sfi 9287	. . . . . . 7 ⊢ ((𝑢 ∈ Fin ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑) → ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))
14	8, 11, 13	syl2anc 585	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))
15		unieq 4920	. . . . . . . . 9 ⊢ (𝑠 = 𝑢 → ∪ 𝑠 = ∪ 𝑢)
16	15	eqeq2d 2744	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → (𝑋 = ∪ 𝑠 ↔ 𝑋 = ∪ 𝑢))
17		feq2 6700	. . . . . . . . . 10 ⊢ (𝑠 = 𝑢 → (𝑓:𝑠⟶𝐴 ↔ 𝑓:𝑢⟶𝐴))
18		raleq 3323	. . . . . . . . . 10 ⊢ (𝑠 = 𝑢 → (∀𝑦 ∈ 𝑠 𝜓 ↔ ∀𝑦 ∈ 𝑢 𝜓))
19	17, 18	anbi12d 632	. . . . . . . . 9 ⊢ (𝑠 = 𝑢 → ((𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓) ↔ (𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓)))
20	19	exbidv 1925	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → (∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓) ↔ ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓)))
21	16, 20	anbi12d 632	. . . . . . 7 ⊢ (𝑠 = 𝑢 → ((𝑋 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓)) ↔ (𝑋 = ∪ 𝑢 ∧ ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))))
22	21	rspcev 3613	. . . . . 6 ⊢ ((𝑢 ∈ (𝒫 𝐽 ∩ Fin) ∧ (𝑋 = ∪ 𝑢 ∧ ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓)))
23	9, 10, 14, 22	syl12anc 836	. . . . 5 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓)))
24	23	ex 414	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) → ((𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓))))
25	4, 24	sylan2b 595	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑢 ∈ (𝒫 𝐽 ∩ Fin)) → ((𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓))))
26	25	rexlimdva 3156	. 2 ⊢ (𝐽 ∈ Comp → (∃𝑢 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓))))
27	1, 3, 26	sylc 65	1 ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝐴 𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ∩ cin 3948   ⊆ wss 3949  𝒫 cpw 4603  ∪ cuni 4909  ⟶wf 6540  ‘cfv 6544  Fincfn 8939  Compccmp 22890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-en 8940  df-fin 8943  df-cmp 22891

This theorem is referenced by:  txtube  23144  txcmplem1  23145  txcmplem2  23146  xkococnlem  23163  cnheibor  24471  heicant  36523