Theorem cmpcovf 23689

Description: Combine cmpcov 23687 with ac6sfi 9259 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 488	. 2 ⊢ ((𝐽 ∈ Comp ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝐽 (𝑥 ∈ 𝑦 ∧ ∃𝑧 ∈ 𝐴 𝜑)) → 𝐽 ∈ Comp)
2		iscmp.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
3	2	cmpcov2 23688	. 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 9327	. . . 4 ⊢ (𝑢 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin))
5		simplrl 789	. . . . . . . 8 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ⊆ 𝐽)
6		velpw 4562	. . . . . . . 8 ⊢ (𝑢 ∈ 𝒫 𝐽 ↔ 𝑢 ⊆ 𝐽)
7	5, 6	sylibr 237	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ 𝒫 𝐽)
8		simplrr 790	. . . . . . 7 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ Fin)
9	7, 8	elind 4146	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑢 ∈ (𝒫 𝐽 ∩ Fin))
10		simprl 783	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → 𝑋 = ∪ 𝑢)
11		simprr 785	. . . . . . 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 9259	. . . . . . 7 ⊢ ((𝑢 ∈ Fin ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑) → ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))
14	8, 11, 13	syl2anc 596	. . . . . 6 ⊢ (((𝐽 ∈ Comp ∧ (𝑢 ⊆ 𝐽 ∧ 𝑢 ∈ Fin)) ∧ (𝑋 = ∪ 𝑢 ∧ ∀𝑦 ∈ 𝑢 ∃𝑧 ∈ 𝐴 𝜑)) → ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))
15		unieq 4878	. . . . . . . . 9 ⊢ (𝑠 = 𝑢 → ∪ 𝑠 = ∪ 𝑢)
16	15	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → (𝑋 = ∪ 𝑠 ↔ 𝑋 = ∪ 𝑢))
17		feq2 6680	. . . . . . . . . 10 ⊢ (𝑠 = 𝑢 → (𝑓:𝑠⟶𝐴 ↔ 𝑓:𝑢⟶𝐴))
18		raleq 3317	. . . . . . . . . 10 ⊢ (𝑠 = 𝑢 → (∀𝑦 ∈ 𝑠 𝜓 ↔ ∀𝑦 ∈ 𝑢 𝜓))
19	17, 18	anbi12d 644	. . . . . . . . 9 ⊢ (𝑠 = 𝑢 → ((𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓) ↔ (𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓)))
20	19	exbidv 1954	. . . . . . . 8 ⊢ (𝑠 = 𝑢 → (∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓) ↔ ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓)))
21	16, 20	anbi12d 644	. . . . . . 7 ⊢ (𝑠 = 𝑢 → ((𝑋 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶𝐴 ∧ ∀𝑦 ∈ 𝑠 𝜓)) ↔ (𝑋 = ∪ 𝑢 ∧ ∃𝑓(𝑓:𝑢⟶𝐴 ∧ ∀𝑦 ∈ 𝑢 𝜓))))
22	21	rspcev 3577	. . . . . 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 850	. . . . 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 418	. . . 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 606	. . 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 3164	. 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 66	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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  ⟶wf 6527  ‘cfv 6531  Fincfn 8957  Compccmp 23684

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-om 7867  df-en 8958  df-fin 8961  df-cmp 23685

This theorem is used by:  txtube  23939  txcmplem1  23940  txcmplem2  23941  xkococnlem  23958  cnheibor  25256  heicant  38541