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 = (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 23216	. 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 9350	. . . 4 ⊢ (𝑢 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin))
5		simplrl 774	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ⊆ 𝐽)
6		velpw 4599	. . . . . . . 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 775	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ Fin)
9	7, 8	elind 4186	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ (𝒫 𝐽 ∩ Fin))
10		simprl 768	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑋 = ∪ 𝑢)
11		simprr 770	. . . . . . 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 9283	. . . . . . 7 ⊢ ((𝑢 ∈ Fin ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑) → ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))
14	8, 11, 13	syl2anc 583	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))
15		unieq 4910	. . . . . . . . 9 ⊢ (𝑠 = 𝑢 → ∪ 𝑠 = ∪ 𝑢)
16	15	eqeq2d 2735	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → (𝑋 = ∪ 𝑠 ↔ 𝑋 = ∪ 𝑢))
17		feq2 6689	. . . . . . . . . 10 ⊢ (𝑠 = 𝑢 → (𝑓:𝑠⟶𝐴 ↔ 𝑓:𝑢⟶𝐴))
18		raleq 3314	. . . . . . . . . 10 ⊢ (𝑠 = 𝑢 → (∀𝑦 ∈ 𝑠 𝜓 ↔ ∀𝑦 ∈ 𝑢 𝜓))
19	17, 18	anbi12d 630	. . . . . . . . 9 ⊢ (𝑠 = 𝑢 → ((𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓) ↔ (𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓)))
20	19	exbidv 1916	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → (∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓) ↔ ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓)))
21	16, 20	anbi12d 630	. . . . . . 7 ⊢ (𝑠 = 𝑢 → ((𝑋 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓)) ↔ (𝑋 = ∪ 𝑢 ∧ ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))))
22	21	rspcev 3604	. . . . . 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 834	. . . . 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 593	. . 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 3147	. 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 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