Theorem cmpcovf 21565

Description: Combine cmpcov 21563 with ac6sfi 8473 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 476	. 2 ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝐴 𝜑)) → 𝐽 ∈ Comp)
2		iscmp.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
3	2	cmpcov2 21564	. 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 8537	. . . 4 ⊢ (𝑢 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin))
5		simplrl 795	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ⊆ 𝐽)
6		selpw 4385	. . . . . . . 8 ⊢ (𝑢 ∈ 𝒫 𝐽 ↔ 𝑢 ⊆ 𝐽)
7	5, 6	sylibr 226	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ 𝒫 𝐽)
8		simplrr 796	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ Fin)
9	7, 8	elind 4025	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ (𝒫 𝐽 ∩ Fin))
10		simprl 787	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑋 = ∪ 𝑢)
11		simprr 789	. . . . . . 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 8473	. . . . . . 7 ⊢ ((𝑢 ∈ Fin ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑) → ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))
14	8, 11, 13	syl2anc 579	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))
15		unieq 4666	. . . . . . . . 9 ⊢ (𝑠 = 𝑢 → ∪ 𝑠 = ∪ 𝑢)
16	15	eqeq2d 2835	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → (𝑋 = ∪ 𝑠 ↔ 𝑋 = ∪ 𝑢))
17		feq2 6260	. . . . . . . . . 10 ⊢ (𝑠 = 𝑢 → (𝑓:𝑠⟶𝐴 ↔ 𝑓:𝑢⟶𝐴))
18		raleq 3350	. . . . . . . . . 10 ⊢ (𝑠 = 𝑢 → (∀𝑦 ∈ 𝑠 𝜓 ↔ ∀𝑦 ∈ 𝑢 𝜓))
19	17, 18	anbi12d 624	. . . . . . . . 9 ⊢ (𝑠 = 𝑢 → ((𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓) ↔ (𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓)))
20	19	exbidv 2020	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → (∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓) ↔ ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓)))
21	16, 20	anbi12d 624	. . . . . . 7 ⊢ (𝑠 = 𝑢 → ((𝑋 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓)) ↔ (𝑋 = ∪ 𝑢 ∧ ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))))
22	21	rspcev 3526	. . . . . 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 870	. . . . 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 403	. . . 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 587	. . 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 3240	. 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 198   ∧ wa 386   = wceq 1656  ∃wex 1878   ∈ wcel 2164  ∀wral 3117  ∃wrex 3118   ∩ cin 3797   ⊆ wss 3798  𝒫 cpw 4378  ∪ cuni 4658  ⟶wf 6119  ‘cfv 6123  Fincfn 8222  Compccmp 21560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-om 7327  df-1o 7826  df-er 8009  df-en 8223  df-fin 8226  df-cmp 21561

This theorem is referenced by:  txtube  21814  txcmplem1  21815  txcmplem2  21816  xkococnlem  21833  cnheibor  23124  heicant  33981