Theorem cmpcovf 23399

Description: Combine cmpcov 23397 with ac6sfi 9320 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 482	. 2 ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝐴 𝜑)) → 𝐽 ∈ Comp)
2		iscmp.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
3	2	cmpcov2 23398	. 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 9394	. . . 4 ⊢ (𝑢 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin))
5		simplrl 777	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ⊆ 𝐽)
6		velpw 4605	. . . . . . . 8 ⊢ (𝑢 ∈ 𝒫 𝐽 ↔ 𝑢 ⊆ 𝐽)
7	5, 6	sylibr 234	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ 𝒫 𝐽)
8		simplrr 778	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ Fin)
9	7, 8	elind 4200	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ (𝒫 𝐽 ∩ Fin))
10		simprl 771	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑋 = ∪ 𝑢)
11		simprr 773	. . . . . . 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 9320	. . . . . . 7 ⊢ ((𝑢 ∈ Fin ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑) → ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))
14	8, 11, 13	syl2anc 584	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))
15		unieq 4918	. . . . . . . . 9 ⊢ (𝑠 = 𝑢 → ∪ 𝑠 = ∪ 𝑢)
16	15	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → (𝑋 = ∪ 𝑠 ↔ 𝑋 = ∪ 𝑢))
17		feq2 6717	. . . . . . . . . 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 1921	. . . . . . . 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 3622	. . . . . 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 837	. . . . 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 412	. . . 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 594	. . 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 3155	. 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 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907  ⟶wf 6557  ‘cfv 6561  Fincfn 8985  Compccmp 23394

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-en 8986  df-fin 8989  df-cmp 23395

This theorem is referenced by:  txtube  23648  txcmplem1  23649  txcmplem2  23650  xkococnlem  23667  cnheibor  24987  heicant  37662