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Theorem cmpcov2 21103
Description: Rewrite cmpcov 21102 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 3573 . . . . 5 (𝑋 {𝑦𝐽𝜑} ↔ ∀𝑥𝑋 𝑥 {𝑦𝐽𝜑})
2 elunirab 4414 . . . . . 6 (𝑥 {𝑦𝐽𝜑} ↔ ∃𝑦𝐽 (𝑥𝑦𝜑))
32ralbii 2974 . . . . 5 (∀𝑥𝑋 𝑥 {𝑦𝐽𝜑} ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑))
41, 3sylbbr 226 . . . 4 (∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑) → 𝑋 {𝑦𝐽𝜑})
5 ssrab2 3666 . . . . . . 7 {𝑦𝐽𝜑} ⊆ 𝐽
65unissi 4427 . . . . . 6 {𝑦𝐽𝜑} ⊆ 𝐽
7 iscmp.1 . . . . . 6 𝑋 = 𝐽
86, 7sseqtr4i 3617 . . . . 5 {𝑦𝐽𝜑} ⊆ 𝑋
98a1i 11 . . . 4 (∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑) → {𝑦𝐽𝜑} ⊆ 𝑋)
104, 9eqssd 3600 . . 3 (∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑) → 𝑋 = {𝑦𝐽𝜑})
117cmpcov 21102 . . . 4 ((𝐽 ∈ Comp ∧ {𝑦𝐽𝜑} ⊆ 𝐽𝑋 = {𝑦𝐽𝜑}) → ∃𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin)𝑋 = 𝑠)
125, 11mp3an2 1409 . . 3 ((𝐽 ∈ Comp ∧ 𝑋 = {𝑦𝐽𝜑}) → ∃𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin)𝑋 = 𝑠)
1310, 12sylan2 491 . 2 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑)) → ∃𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin)𝑋 = 𝑠)
14 ssrab 3659 . . . . . . . 8 (𝑠 ⊆ {𝑦𝐽𝜑} ↔ (𝑠𝐽 ∧ ∀𝑦𝑠 𝜑))
1514anbi1i 730 . . . . . . 7 ((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑋 = 𝑠) ↔ ((𝑠𝐽 ∧ ∀𝑦𝑠 𝜑) ∧ 𝑋 = 𝑠))
16 an32 838 . . . . . . 7 (((𝑠𝐽 ∧ ∀𝑦𝑠 𝜑) ∧ 𝑋 = 𝑠) ↔ ((𝑠𝐽𝑋 = 𝑠) ∧ ∀𝑦𝑠 𝜑))
17 anass 680 . . . . . . 7 (((𝑠𝐽𝑋 = 𝑠) ∧ ∀𝑦𝑠 𝜑) ↔ (𝑠𝐽 ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)))
1815, 16, 173bitri 286 . . . . . 6 ((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑋 = 𝑠) ↔ (𝑠𝐽 ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)))
1918anbi1i 730 . . . . 5 (((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑋 = 𝑠) ∧ 𝑠 ∈ Fin) ↔ ((𝑠𝐽 ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)) ∧ 𝑠 ∈ Fin))
20 an32 838 . . . . 5 (((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑠 ∈ Fin) ∧ 𝑋 = 𝑠) ↔ ((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑋 = 𝑠) ∧ 𝑠 ∈ Fin))
21 an32 838 . . . . 5 (((𝑠𝐽𝑠 ∈ Fin) ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)) ↔ ((𝑠𝐽 ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)) ∧ 𝑠 ∈ Fin))
2219, 20, 213bitr4i 292 . . . 4 (((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑠 ∈ Fin) ∧ 𝑋 = 𝑠) ↔ ((𝑠𝐽𝑠 ∈ Fin) ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)))
23 elfpw 8212 . . . . 5 (𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin) ↔ (𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑠 ∈ Fin))
2423anbi1i 730 . . . 4 ((𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin) ∧ 𝑋 = 𝑠) ↔ ((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑠 ∈ Fin) ∧ 𝑋 = 𝑠))
25 elfpw 8212 . . . . 5 (𝑠 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑠𝐽𝑠 ∈ Fin))
2625anbi1i 730 . . . 4 ((𝑠 ∈ (𝒫 𝐽 ∩ Fin) ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)) ↔ ((𝑠𝐽𝑠 ∈ Fin) ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)))
2722, 24, 263bitr4i 292 . . 3 ((𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin) ∧ 𝑋 = 𝑠) ↔ (𝑠 ∈ (𝒫 𝐽 ∩ Fin) ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)))
2827rexbii2 3032 . 2 (∃𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin)𝑋 = 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑))
2913, 28sylib 208 1 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  {crab 2911  cin 3554  wss 3555  𝒫 cpw 4130   cuni 4402  Fincfn 7899  Compccmp 21099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-in 3562  df-ss 3569  df-pw 4132  df-uni 4403  df-cmp 21100
This theorem is referenced by:  cmpcovf  21104  bwth  21123  locfincmp  21239
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