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Theorem cmpcov2 23414
Description: Rewrite cmpcov 23413 for the cover {𝑦𝐽𝜑}. (Contributed by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
iscmp.1 𝑋 = 𝐽
Assertion
Ref Expression
cmpcov2 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑))
Distinct variable groups:   𝑥,𝑠,𝑦,𝐽   𝜑,𝑠,𝑥   𝑥,𝑋
Allowed substitution hints:   𝜑(𝑦)   𝑋(𝑦,𝑠)

Proof of Theorem cmpcov2
StepHypRef Expression
1 dfss3 3984 . . . . 5 (𝑋 {𝑦𝐽𝜑} ↔ ∀𝑥𝑋 𝑥 {𝑦𝐽𝜑})
2 elunirab 4927 . . . . . 6 (𝑥 {𝑦𝐽𝜑} ↔ ∃𝑦𝐽 (𝑥𝑦𝜑))
32ralbii 3091 . . . . 5 (∀𝑥𝑋 𝑥 {𝑦𝐽𝜑} ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑))
41, 3sylbbr 236 . . . 4 (∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑) → 𝑋 {𝑦𝐽𝜑})
5 ssrab2 4090 . . . . . . 7 {𝑦𝐽𝜑} ⊆ 𝐽
65unissi 4921 . . . . . 6 {𝑦𝐽𝜑} ⊆ 𝐽
7 iscmp.1 . . . . . 6 𝑋 = 𝐽
86, 7sseqtrri 4033 . . . . 5 {𝑦𝐽𝜑} ⊆ 𝑋
98a1i 11 . . . 4 (∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑) → {𝑦𝐽𝜑} ⊆ 𝑋)
104, 9eqssd 4013 . . 3 (∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑) → 𝑋 = {𝑦𝐽𝜑})
117cmpcov 23413 . . . 4 ((𝐽 ∈ Comp ∧ {𝑦𝐽𝜑} ⊆ 𝐽𝑋 = {𝑦𝐽𝜑}) → ∃𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin)𝑋 = 𝑠)
125, 11mp3an2 1448 . . 3 ((𝐽 ∈ Comp ∧ 𝑋 = {𝑦𝐽𝜑}) → ∃𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin)𝑋 = 𝑠)
1310, 12sylan2 593 . 2 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑)) → ∃𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin)𝑋 = 𝑠)
14 ssrab 4083 . . . . . . . 8 (𝑠 ⊆ {𝑦𝐽𝜑} ↔ (𝑠𝐽 ∧ ∀𝑦𝑠 𝜑))
1514anbi1i 624 . . . . . . 7 ((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑋 = 𝑠) ↔ ((𝑠𝐽 ∧ ∀𝑦𝑠 𝜑) ∧ 𝑋 = 𝑠))
16 an32 646 . . . . . . 7 (((𝑠𝐽 ∧ ∀𝑦𝑠 𝜑) ∧ 𝑋 = 𝑠) ↔ ((𝑠𝐽𝑋 = 𝑠) ∧ ∀𝑦𝑠 𝜑))
17 anass 468 . . . . . . 7 (((𝑠𝐽𝑋 = 𝑠) ∧ ∀𝑦𝑠 𝜑) ↔ (𝑠𝐽 ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)))
1815, 16, 173bitri 297 . . . . . 6 ((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑋 = 𝑠) ↔ (𝑠𝐽 ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)))
1918anbi1i 624 . . . . 5 (((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑋 = 𝑠) ∧ 𝑠 ∈ Fin) ↔ ((𝑠𝐽 ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)) ∧ 𝑠 ∈ Fin))
20 an32 646 . . . . 5 (((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑠 ∈ Fin) ∧ 𝑋 = 𝑠) ↔ ((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑋 = 𝑠) ∧ 𝑠 ∈ Fin))
21 an32 646 . . . . 5 (((𝑠𝐽𝑠 ∈ Fin) ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)) ↔ ((𝑠𝐽 ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)) ∧ 𝑠 ∈ Fin))
2219, 20, 213bitr4i 303 . . . 4 (((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑠 ∈ Fin) ∧ 𝑋 = 𝑠) ↔ ((𝑠𝐽𝑠 ∈ Fin) ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)))
23 elfpw 9392 . . . . 5 (𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin) ↔ (𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑠 ∈ Fin))
2423anbi1i 624 . . . 4 ((𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin) ∧ 𝑋 = 𝑠) ↔ ((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑠 ∈ Fin) ∧ 𝑋 = 𝑠))
25 elfpw 9392 . . . . 5 (𝑠 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑠𝐽𝑠 ∈ Fin))
2625anbi1i 624 . . . 4 ((𝑠 ∈ (𝒫 𝐽 ∩ Fin) ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)) ↔ ((𝑠𝐽𝑠 ∈ Fin) ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)))
2722, 24, 263bitr4i 303 . . 3 ((𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin) ∧ 𝑋 = 𝑠) ↔ (𝑠 ∈ (𝒫 𝐽 ∩ Fin) ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)))
2827rexbii2 3088 . 2 (∃𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin)𝑋 = 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑))
2913, 28sylib 218 1 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912  Fincfn 8984  Compccmp 23410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-in 3970  df-ss 3980  df-pw 4607  df-uni 4913  df-cmp 23411
This theorem is referenced by:  cmpcovf  23415  bwth  23434  locfincmp  23550
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