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Theorem cover2 34542
Description: Two ways of expressing the statement "there is a cover of 𝐴 by elements of 𝐵 such that for each set in the cover, 𝜑". Note that 𝜑 and 𝑥 must be distinct. (Contributed by Jeff Madsen, 20-Jun-2010.)
Hypotheses
Ref Expression
cover2.1 𝐵 ∈ V
cover2.2 𝐴 = 𝐵
Assertion
Ref Expression
cover2 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑))
Distinct variable groups:   𝜑,𝑥,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐴,𝑧
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)

Proof of Theorem cover2
StepHypRef Expression
1 cover2.1 . . . 4 𝐵 ∈ V
2 ssrab2 3983 . . . 4 {𝑦𝐵𝜑} ⊆ 𝐵
31, 2elpwi2 5147 . . 3 {𝑦𝐵𝜑} ∈ 𝒫 𝐵
4 nfra1 3188 . . . . 5 𝑥𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑)
52unissi 4774 . . . . . . . 8 {𝑦𝐵𝜑} ⊆ 𝐵
65sseli 3891 . . . . . . 7 (𝑥 {𝑦𝐵𝜑} → 𝑥 𝐵)
7 cover2.2 . . . . . . 7 𝐴 = 𝐵
86, 7syl6eleqr 2896 . . . . . 6 (𝑥 {𝑦𝐵𝜑} → 𝑥𝐴)
9 rsp 3174 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → (𝑥𝐴 → ∃𝑦𝐵 (𝑥𝑦𝜑)))
10 elunirab 4763 . . . . . . 7 (𝑥 {𝑦𝐵𝜑} ↔ ∃𝑦𝐵 (𝑥𝑦𝜑))
119, 10syl6ibr 253 . . . . . 6 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → (𝑥𝐴𝑥 {𝑦𝐵𝜑}))
128, 11impbid2 227 . . . . 5 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → (𝑥 {𝑦𝐵𝜑} ↔ 𝑥𝐴))
134, 12alrimi 2180 . . . 4 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → ∀𝑥(𝑥 {𝑦𝐵𝜑} ↔ 𝑥𝐴))
14 dfcleq 2791 . . . 4 ( {𝑦𝐵𝜑} = 𝐴 ↔ ∀𝑥(𝑥 {𝑦𝐵𝜑} ↔ 𝑥𝐴))
1513, 14sylibr 235 . . 3 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → {𝑦𝐵𝜑} = 𝐴)
16 nfrab1 3346 . . . . . . 7 𝑦{𝑦𝐵𝜑}
1716nfeq2 2966 . . . . . 6 𝑦 𝑧 = {𝑦𝐵𝜑}
18 eleq2 2873 . . . . . . 7 (𝑧 = {𝑦𝐵𝜑} → (𝑦𝑧𝑦 ∈ {𝑦𝐵𝜑}))
19 rabid 3339 . . . . . . . 8 (𝑦 ∈ {𝑦𝐵𝜑} ↔ (𝑦𝐵𝜑))
2019simprbi 497 . . . . . . 7 (𝑦 ∈ {𝑦𝐵𝜑} → 𝜑)
2118, 20syl6bi 254 . . . . . 6 (𝑧 = {𝑦𝐵𝜑} → (𝑦𝑧𝜑))
2217, 21ralrimi 3185 . . . . 5 (𝑧 = {𝑦𝐵𝜑} → ∀𝑦𝑧 𝜑)
23 unieq 4759 . . . . . . 7 (𝑧 = {𝑦𝐵𝜑} → 𝑧 = {𝑦𝐵𝜑})
2423eqeq1d 2799 . . . . . 6 (𝑧 = {𝑦𝐵𝜑} → ( 𝑧 = 𝐴 {𝑦𝐵𝜑} = 𝐴))
2524anbi1d 629 . . . . 5 (𝑧 = {𝑦𝐵𝜑} → (( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑) ↔ ( {𝑦𝐵𝜑} = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
2622, 25mpbiran2d 704 . . . 4 (𝑧 = {𝑦𝐵𝜑} → (( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑) ↔ {𝑦𝐵𝜑} = 𝐴))
2726rspcev 3561 . . 3 (({𝑦𝐵𝜑} ∈ 𝒫 𝐵 {𝑦𝐵𝜑} = 𝐴) → ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑))
283, 15, 27sylancr 587 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑))
29 elpwi 4469 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
30 r19.29r 3221 . . . . . . . . . . 11 ((∃𝑦𝑧 𝑥𝑦 ∧ ∀𝑦𝑧 𝜑) → ∃𝑦𝑧 (𝑥𝑦𝜑))
3130expcom 414 . . . . . . . . . 10 (∀𝑦𝑧 𝜑 → (∃𝑦𝑧 𝑥𝑦 → ∃𝑦𝑧 (𝑥𝑦𝜑)))
32 ssrexv 3961 . . . . . . . . . 10 (𝑧𝐵 → (∃𝑦𝑧 (𝑥𝑦𝜑) → ∃𝑦𝐵 (𝑥𝑦𝜑)))
3331, 32sylan9r 509 . . . . . . . . 9 ((𝑧𝐵 ∧ ∀𝑦𝑧 𝜑) → (∃𝑦𝑧 𝑥𝑦 → ∃𝑦𝐵 (𝑥𝑦𝜑)))
3429, 33sylan 580 . . . . . . . 8 ((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦𝑧 𝜑) → (∃𝑦𝑧 𝑥𝑦 → ∃𝑦𝐵 (𝑥𝑦𝜑)))
35 eleq2 2873 . . . . . . . . . 10 ( 𝑧 = 𝐴 → (𝑥 𝑧𝑥𝐴))
3635biimpar 478 . . . . . . . . 9 (( 𝑧 = 𝐴𝑥𝐴) → 𝑥 𝑧)
37 eluni2 4755 . . . . . . . . 9 (𝑥 𝑧 ↔ ∃𝑦𝑧 𝑥𝑦)
3836, 37sylib 219 . . . . . . . 8 (( 𝑧 = 𝐴𝑥𝐴) → ∃𝑦𝑧 𝑥𝑦)
3934, 38impel 506 . . . . . . 7 (((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦𝑧 𝜑) ∧ ( 𝑧 = 𝐴𝑥𝐴)) → ∃𝑦𝐵 (𝑥𝑦𝜑))
4039anassrs 468 . . . . . 6 ((((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦𝑧 𝜑) ∧ 𝑧 = 𝐴) ∧ 𝑥𝐴) → ∃𝑦𝐵 (𝑥𝑦𝜑))
4140ralrimiva 3151 . . . . 5 (((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦𝑧 𝜑) ∧ 𝑧 = 𝐴) → ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑))
4241anasss 467 . . . 4 ((𝑧 ∈ 𝒫 𝐵 ∧ (∀𝑦𝑧 𝜑 𝑧 = 𝐴)) → ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑))
4342ancom2s 646 . . 3 ((𝑧 ∈ 𝒫 𝐵 ∧ ( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)) → ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑))
4443rexlimiva 3246 . 2 (∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑) → ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑))
4528, 44impbii 210 1 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1523   = wceq 1525  wcel 2083  wral 3107  wrex 3108  {crab 3111  Vcvv 3440  wss 3865  𝒫 cpw 4459   cuni 4751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771  ax-sep 5101
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-in 3872  df-ss 3880  df-pw 4461  df-uni 4752
This theorem is referenced by:  cover2g  34543
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