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Theorem cover2 33863
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 ssrab2 3846 . . . 4 {𝑦𝐵𝜑} ⊆ 𝐵
2 cover2.1 . . . . 5 𝐵 ∈ V
32elpw2 4985 . . . 4 ({𝑦𝐵𝜑} ∈ 𝒫 𝐵 ↔ {𝑦𝐵𝜑} ⊆ 𝐵)
41, 3mpbir 222 . . 3 {𝑦𝐵𝜑} ∈ 𝒫 𝐵
5 nfra1 3087 . . . . 5 𝑥𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑)
61unissi 4618 . . . . . . . 8 {𝑦𝐵𝜑} ⊆ 𝐵
76sseli 3756 . . . . . . 7 (𝑥 {𝑦𝐵𝜑} → 𝑥 𝐵)
8 cover2.2 . . . . . . 7 𝐴 = 𝐵
97, 8syl6eleqr 2854 . . . . . 6 (𝑥 {𝑦𝐵𝜑} → 𝑥𝐴)
10 rsp 3075 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → (𝑥𝐴 → ∃𝑦𝐵 (𝑥𝑦𝜑)))
11 elunirab 4605 . . . . . . 7 (𝑥 {𝑦𝐵𝜑} ↔ ∃𝑦𝐵 (𝑥𝑦𝜑))
1210, 11syl6ibr 243 . . . . . 6 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → (𝑥𝐴𝑥 {𝑦𝐵𝜑}))
139, 12impbid2 217 . . . . 5 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → (𝑥 {𝑦𝐵𝜑} ↔ 𝑥𝐴))
145, 13alrimi 2246 . . . 4 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → ∀𝑥(𝑥 {𝑦𝐵𝜑} ↔ 𝑥𝐴))
15 dfcleq 2758 . . . 4 ( {𝑦𝐵𝜑} = 𝐴 ↔ ∀𝑥(𝑥 {𝑦𝐵𝜑} ↔ 𝑥𝐴))
1614, 15sylibr 225 . . 3 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → {𝑦𝐵𝜑} = 𝐴)
17 unieq 4601 . . . . . . 7 (𝑧 = {𝑦𝐵𝜑} → 𝑧 = {𝑦𝐵𝜑})
1817eqeq1d 2766 . . . . . 6 (𝑧 = {𝑦𝐵𝜑} → ( 𝑧 = 𝐴 {𝑦𝐵𝜑} = 𝐴))
1918anbi1d 623 . . . . 5 (𝑧 = {𝑦𝐵𝜑} → (( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑) ↔ ( {𝑦𝐵𝜑} = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
20 nfrab1 3269 . . . . . . . 8 𝑦{𝑦𝐵𝜑}
2120nfeq2 2922 . . . . . . 7 𝑦 𝑧 = {𝑦𝐵𝜑}
22 eleq2 2832 . . . . . . . 8 (𝑧 = {𝑦𝐵𝜑} → (𝑦𝑧𝑦 ∈ {𝑦𝐵𝜑}))
23 rabid 3262 . . . . . . . . 9 (𝑦 ∈ {𝑦𝐵𝜑} ↔ (𝑦𝐵𝜑))
2423simprbi 490 . . . . . . . 8 (𝑦 ∈ {𝑦𝐵𝜑} → 𝜑)
2522, 24syl6bi 244 . . . . . . 7 (𝑧 = {𝑦𝐵𝜑} → (𝑦𝑧𝜑))
2621, 25ralrimi 3103 . . . . . 6 (𝑧 = {𝑦𝐵𝜑} → ∀𝑦𝑧 𝜑)
2726biantrud 527 . . . . 5 (𝑧 = {𝑦𝐵𝜑} → ( {𝑦𝐵𝜑} = 𝐴 ↔ ( {𝑦𝐵𝜑} = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
2819, 27bitr4d 273 . . . 4 (𝑧 = {𝑦𝐵𝜑} → (( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑) ↔ {𝑦𝐵𝜑} = 𝐴))
2928rspcev 3460 . . 3 (({𝑦𝐵𝜑} ∈ 𝒫 𝐵 {𝑦𝐵𝜑} = 𝐴) → ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑))
304, 16, 29sylancr 581 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑))
31 elpwi 4324 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
32 r19.29r 3219 . . . . . . . . . . 11 ((∃𝑦𝑧 𝑥𝑦 ∧ ∀𝑦𝑧 𝜑) → ∃𝑦𝑧 (𝑥𝑦𝜑))
3332expcom 402 . . . . . . . . . 10 (∀𝑦𝑧 𝜑 → (∃𝑦𝑧 𝑥𝑦 → ∃𝑦𝑧 (𝑥𝑦𝜑)))
34 ssrexv 3826 . . . . . . . . . 10 (𝑧𝐵 → (∃𝑦𝑧 (𝑥𝑦𝜑) → ∃𝑦𝐵 (𝑥𝑦𝜑)))
3533, 34sylan9r 504 . . . . . . . . 9 ((𝑧𝐵 ∧ ∀𝑦𝑧 𝜑) → (∃𝑦𝑧 𝑥𝑦 → ∃𝑦𝐵 (𝑥𝑦𝜑)))
3631, 35sylan 575 . . . . . . . 8 ((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦𝑧 𝜑) → (∃𝑦𝑧 𝑥𝑦 → ∃𝑦𝐵 (𝑥𝑦𝜑)))
37 eleq2 2832 . . . . . . . . . 10 ( 𝑧 = 𝐴 → (𝑥 𝑧𝑥𝐴))
3837biimpar 469 . . . . . . . . 9 (( 𝑧 = 𝐴𝑥𝐴) → 𝑥 𝑧)
39 eluni2 4597 . . . . . . . . 9 (𝑥 𝑧 ↔ ∃𝑦𝑧 𝑥𝑦)
4038, 39sylib 209 . . . . . . . 8 (( 𝑧 = 𝐴𝑥𝐴) → ∃𝑦𝑧 𝑥𝑦)
4136, 40impel 501 . . . . . . 7 (((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦𝑧 𝜑) ∧ ( 𝑧 = 𝐴𝑥𝐴)) → ∃𝑦𝐵 (𝑥𝑦𝜑))
4241anassrs 459 . . . . . 6 ((((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦𝑧 𝜑) ∧ 𝑧 = 𝐴) ∧ 𝑥𝐴) → ∃𝑦𝐵 (𝑥𝑦𝜑))
4342ralrimiva 3112 . . . . 5 (((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦𝑧 𝜑) ∧ 𝑧 = 𝐴) → ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑))
4443anasss 458 . . . 4 ((𝑧 ∈ 𝒫 𝐵 ∧ (∀𝑦𝑧 𝜑 𝑧 = 𝐴)) → ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑))
4544ancom2s 640 . . 3 ((𝑧 ∈ 𝒫 𝐵 ∧ ( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)) → ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑))
4645rexlimiva 3174 . 2 (∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑) → ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑))
4730, 46impbii 200 1 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wcel 2155  wral 3054  wrex 3055  {crab 3058  Vcvv 3349  wss 3731  𝒫 cpw 4314   cuni 4593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-in 3738  df-ss 3745  df-pw 4316  df-uni 4594
This theorem is referenced by:  cover2g  33864
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