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Theorem cover2 33821
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 3828 . . . 4 {𝑦𝐵𝜑} ⊆ 𝐵
2 cover2.1 . . . . 5 𝐵 ∈ V
32elpw2 4977 . . . 4 ({𝑦𝐵𝜑} ∈ 𝒫 𝐵 ↔ {𝑦𝐵𝜑} ⊆ 𝐵)
41, 3mpbir 221 . . 3 {𝑦𝐵𝜑} ∈ 𝒫 𝐵
5 nfra1 3079 . . . . 5 𝑥𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑)
61unissi 4613 . . . . . . . 8 {𝑦𝐵𝜑} ⊆ 𝐵
76sseli 3740 . . . . . . 7 (𝑥 {𝑦𝐵𝜑} → 𝑥 𝐵)
8 cover2.2 . . . . . . 7 𝐴 = 𝐵
97, 8syl6eleqr 2850 . . . . . 6 (𝑥 {𝑦𝐵𝜑} → 𝑥𝐴)
10 rsp 3067 . . . . . . 7 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → (𝑥𝐴 → ∃𝑦𝐵 (𝑥𝑦𝜑)))
11 elunirab 4600 . . . . . . 7 (𝑥 {𝑦𝐵𝜑} ↔ ∃𝑦𝐵 (𝑥𝑦𝜑))
1210, 11syl6ibr 242 . . . . . 6 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → (𝑥𝐴𝑥 {𝑦𝐵𝜑}))
139, 12impbid2 216 . . . . 5 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → (𝑥 {𝑦𝐵𝜑} ↔ 𝑥𝐴))
145, 13alrimi 2229 . . . 4 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → ∀𝑥(𝑥 {𝑦𝐵𝜑} ↔ 𝑥𝐴))
15 dfcleq 2754 . . . 4 ( {𝑦𝐵𝜑} = 𝐴 ↔ ∀𝑥(𝑥 {𝑦𝐵𝜑} ↔ 𝑥𝐴))
1614, 15sylibr 224 . . 3 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → {𝑦𝐵𝜑} = 𝐴)
17 unieq 4596 . . . . . . 7 (𝑧 = {𝑦𝐵𝜑} → 𝑧 = {𝑦𝐵𝜑})
1817eqeq1d 2762 . . . . . 6 (𝑧 = {𝑦𝐵𝜑} → ( 𝑧 = 𝐴 {𝑦𝐵𝜑} = 𝐴))
1918anbi1d 743 . . . . 5 (𝑧 = {𝑦𝐵𝜑} → (( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑) ↔ ( {𝑦𝐵𝜑} = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
20 nfrab1 3261 . . . . . . . 8 𝑦{𝑦𝐵𝜑}
2120nfeq2 2918 . . . . . . 7 𝑦 𝑧 = {𝑦𝐵𝜑}
22 eleq2 2828 . . . . . . . 8 (𝑧 = {𝑦𝐵𝜑} → (𝑦𝑧𝑦 ∈ {𝑦𝐵𝜑}))
23 rabid 3254 . . . . . . . . 9 (𝑦 ∈ {𝑦𝐵𝜑} ↔ (𝑦𝐵𝜑))
2423simprbi 483 . . . . . . . 8 (𝑦 ∈ {𝑦𝐵𝜑} → 𝜑)
2522, 24syl6bi 243 . . . . . . 7 (𝑧 = {𝑦𝐵𝜑} → (𝑦𝑧𝜑))
2621, 25ralrimi 3095 . . . . . 6 (𝑧 = {𝑦𝐵𝜑} → ∀𝑦𝑧 𝜑)
2726biantrud 529 . . . . 5 (𝑧 = {𝑦𝐵𝜑} → ( {𝑦𝐵𝜑} = 𝐴 ↔ ( {𝑦𝐵𝜑} = 𝐴 ∧ ∀𝑦𝑧 𝜑)))
2819, 27bitr4d 271 . . . 4 (𝑧 = {𝑦𝐵𝜑} → (( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑) ↔ {𝑦𝐵𝜑} = 𝐴))
2928rspcev 3449 . . 3 (({𝑦𝐵𝜑} ∈ 𝒫 𝐵 {𝑦𝐵𝜑} = 𝐴) → ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑))
304, 16, 29sylancr 698 . 2 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) → ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑))
31 elpwi 4312 . . . . . . . . 9 (𝑧 ∈ 𝒫 𝐵𝑧𝐵)
32 r19.29r 3211 . . . . . . . . . . 11 ((∃𝑦𝑧 𝑥𝑦 ∧ ∀𝑦𝑧 𝜑) → ∃𝑦𝑧 (𝑥𝑦𝜑))
3332expcom 450 . . . . . . . . . 10 (∀𝑦𝑧 𝜑 → (∃𝑦𝑧 𝑥𝑦 → ∃𝑦𝑧 (𝑥𝑦𝜑)))
34 ssrexv 3808 . . . . . . . . . 10 (𝑧𝐵 → (∃𝑦𝑧 (𝑥𝑦𝜑) → ∃𝑦𝐵 (𝑥𝑦𝜑)))
3533, 34sylan9r 693 . . . . . . . . 9 ((𝑧𝐵 ∧ ∀𝑦𝑧 𝜑) → (∃𝑦𝑧 𝑥𝑦 → ∃𝑦𝐵 (𝑥𝑦𝜑)))
3631, 35sylan 489 . . . . . . . 8 ((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦𝑧 𝜑) → (∃𝑦𝑧 𝑥𝑦 → ∃𝑦𝐵 (𝑥𝑦𝜑)))
37 eleq2 2828 . . . . . . . . . 10 ( 𝑧 = 𝐴 → (𝑥 𝑧𝑥𝐴))
3837biimpar 503 . . . . . . . . 9 (( 𝑧 = 𝐴𝑥𝐴) → 𝑥 𝑧)
39 eluni2 4592 . . . . . . . . 9 (𝑥 𝑧 ↔ ∃𝑦𝑧 𝑥𝑦)
4038, 39sylib 208 . . . . . . . 8 (( 𝑧 = 𝐴𝑥𝐴) → ∃𝑦𝑧 𝑥𝑦)
4136, 40impel 486 . . . . . . 7 (((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦𝑧 𝜑) ∧ ( 𝑧 = 𝐴𝑥𝐴)) → ∃𝑦𝐵 (𝑥𝑦𝜑))
4241anassrs 683 . . . . . 6 ((((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦𝑧 𝜑) ∧ 𝑧 = 𝐴) ∧ 𝑥𝐴) → ∃𝑦𝐵 (𝑥𝑦𝜑))
4342ralrimiva 3104 . . . . 5 (((𝑧 ∈ 𝒫 𝐵 ∧ ∀𝑦𝑧 𝜑) ∧ 𝑧 = 𝐴) → ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑))
4443anasss 682 . . . 4 ((𝑧 ∈ 𝒫 𝐵 ∧ (∀𝑦𝑧 𝜑 𝑧 = 𝐴)) → ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑))
4544ancom2s 879 . . 3 ((𝑧 ∈ 𝒫 𝐵 ∧ ( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑)) → ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑))
4645rexlimiva 3166 . 2 (∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑) → ∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑))
4730, 46impbii 199 1 (∀𝑥𝐴𝑦𝐵 (𝑥𝑦𝜑) ↔ ∃𝑧 ∈ 𝒫 𝐵( 𝑧 = 𝐴 ∧ ∀𝑦𝑧 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  wss 3715  𝒫 cpw 4302   cuni 4588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-in 3722  df-ss 3729  df-pw 4304  df-uni 4589
This theorem is referenced by:  cover2g  33822
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