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Theorem cmpcov2 22764
Description: Rewrite cmpcov 22763 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 3936 . . . . 5 (𝑋 {𝑦𝐽𝜑} ↔ ∀𝑥𝑋 𝑥 {𝑦𝐽𝜑})
2 elunirab 4885 . . . . . 6 (𝑥 {𝑦𝐽𝜑} ↔ ∃𝑦𝐽 (𝑥𝑦𝜑))
32ralbii 3093 . . . . 5 (∀𝑥𝑋 𝑥 {𝑦𝐽𝜑} ↔ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑))
41, 3sylbbr 235 . . . 4 (∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑) → 𝑋 {𝑦𝐽𝜑})
5 ssrab2 4041 . . . . . . 7 {𝑦𝐽𝜑} ⊆ 𝐽
65unissi 4878 . . . . . 6 {𝑦𝐽𝜑} ⊆ 𝐽
7 iscmp.1 . . . . . 6 𝑋 = 𝐽
86, 7sseqtrri 3985 . . . . 5 {𝑦𝐽𝜑} ⊆ 𝑋
98a1i 11 . . . 4 (∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑) → {𝑦𝐽𝜑} ⊆ 𝑋)
104, 9eqssd 3965 . . 3 (∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑) → 𝑋 = {𝑦𝐽𝜑})
117cmpcov 22763 . . . 4 ((𝐽 ∈ Comp ∧ {𝑦𝐽𝜑} ⊆ 𝐽𝑋 = {𝑦𝐽𝜑}) → ∃𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin)𝑋 = 𝑠)
125, 11mp3an2 1450 . . 3 ((𝐽 ∈ Comp ∧ 𝑋 = {𝑦𝐽𝜑}) → ∃𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin)𝑋 = 𝑠)
1310, 12sylan2 594 . 2 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑)) → ∃𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin)𝑋 = 𝑠)
14 ssrab 4034 . . . . . . . 8 (𝑠 ⊆ {𝑦𝐽𝜑} ↔ (𝑠𝐽 ∧ ∀𝑦𝑠 𝜑))
1514anbi1i 625 . . . . . . 7 ((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑋 = 𝑠) ↔ ((𝑠𝐽 ∧ ∀𝑦𝑠 𝜑) ∧ 𝑋 = 𝑠))
16 an32 645 . . . . . . 7 (((𝑠𝐽 ∧ ∀𝑦𝑠 𝜑) ∧ 𝑋 = 𝑠) ↔ ((𝑠𝐽𝑋 = 𝑠) ∧ ∀𝑦𝑠 𝜑))
17 anass 470 . . . . . . 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 625 . . . . 5 (((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑋 = 𝑠) ∧ 𝑠 ∈ Fin) ↔ ((𝑠𝐽 ∧ (𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑)) ∧ 𝑠 ∈ Fin))
20 an32 645 . . . . 5 (((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑠 ∈ Fin) ∧ 𝑋 = 𝑠) ↔ ((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑋 = 𝑠) ∧ 𝑠 ∈ Fin))
21 an32 645 . . . . 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 9304 . . . . 5 (𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin) ↔ (𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑠 ∈ Fin))
2423anbi1i 625 . . . 4 ((𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin) ∧ 𝑋 = 𝑠) ↔ ((𝑠 ⊆ {𝑦𝐽𝜑} ∧ 𝑠 ∈ Fin) ∧ 𝑋 = 𝑠))
25 elfpw 9304 . . . . 5 (𝑠 ∈ (𝒫 𝐽 ∩ Fin) ↔ (𝑠𝐽𝑠 ∈ Fin))
2625anbi1i 625 . . . 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 3090 . 2 (∃𝑠 ∈ (𝒫 {𝑦𝐽𝜑} ∩ Fin)𝑋 = 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑))
2913, 28sylib 217 1 ((𝐽 ∈ Comp ∧ ∀𝑥𝑋𝑦𝐽 (𝑥𝑦𝜑)) → ∃𝑠 ∈ (𝒫 𝐽 ∩ Fin)(𝑋 = 𝑠 ∧ ∀𝑦𝑠 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  wrex 3070  {crab 3406  cin 3913  wss 3914  𝒫 cpw 4564   cuni 4869  Fincfn 8889  Compccmp 22760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-in 3921  df-ss 3931  df-pw 4566  df-uni 4870  df-cmp 22761
This theorem is referenced by:  cmpcovf  22765  bwth  22784  locfincmp  22900
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