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Theorem fpwwelem 9651
Description: Lemma for fpwwe 9652. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
fpwwe.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
fpwwe.2 (𝜑𝐴 ∈ V)
Assertion
Ref Expression
fpwwelem (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))))
Distinct variable groups:   𝑥,𝑟,𝐴   𝑦,𝑟,𝐹,𝑥   𝜑,𝑟,𝑥,𝑦   𝑅,𝑟,𝑥,𝑦   𝑋,𝑟,𝑥,𝑦   𝑊,𝑟,𝑥,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem fpwwelem
StepHypRef Expression
1 fpwwe.1 . . . . 5 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
21relopabi 5393 . . . 4 Rel 𝑊
32a1i 11 . . 3 (𝜑 → Rel 𝑊)
4 brrelex12 5304 . . 3 ((Rel 𝑊𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
53, 4sylan 489 . 2 ((𝜑𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
6 fpwwe.2 . . . . 5 (𝜑𝐴 ∈ V)
76adantr 472 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))) → 𝐴 ∈ V)
8 simprll 821 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))) → 𝑋𝐴)
97, 8ssexd 4949 . . 3 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))) → 𝑋 ∈ V)
10 xpexg 7117 . . . . 5 ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑋 × 𝑋) ∈ V)
119, 9, 10syl2anc 696 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))) → (𝑋 × 𝑋) ∈ V)
12 simprlr 822 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))) → 𝑅 ⊆ (𝑋 × 𝑋))
1311, 12ssexd 4949 . . 3 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))) → 𝑅 ∈ V)
149, 13jca 555 . 2 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
15 simpl 474 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → 𝑥 = 𝑋)
1615sseq1d 3765 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑥𝐴𝑋𝐴))
17 simpr 479 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → 𝑟 = 𝑅)
1815sqxpeqd 5290 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
1917, 18sseq12d 3767 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
2016, 19anbi12d 749 . . . 4 ((𝑥 = 𝑋𝑟 = 𝑅) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋))))
21 weeq2 5247 . . . . . 6 (𝑥 = 𝑋 → (𝑟 We 𝑥𝑟 We 𝑋))
22 weeq1 5246 . . . . . 6 (𝑟 = 𝑅 → (𝑟 We 𝑋𝑅 We 𝑋))
2321, 22sylan9bb 738 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 We 𝑥𝑅 We 𝑋))
2417cnveqd 5445 . . . . . . . . 9 ((𝑥 = 𝑋𝑟 = 𝑅) → 𝑟 = 𝑅)
2524imaeq1d 5615 . . . . . . . 8 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 “ {𝑦}) = (𝑅 “ {𝑦}))
2625fveq2d 6348 . . . . . . 7 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝐹‘(𝑟 “ {𝑦})) = (𝐹‘(𝑅 “ {𝑦})))
2726eqeq1d 2754 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → ((𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ (𝐹‘(𝑅 “ {𝑦})) = 𝑦))
2815, 27raleqbidv 3283 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))
2923, 28anbi12d 749 . . . 4 ((𝑥 = 𝑋𝑟 = 𝑅) → ((𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦) ↔ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦)))
3020, 29anbi12d 749 . . 3 ((𝑥 = 𝑋𝑟 = 𝑅) → (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦)) ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))))
3130, 1brabga 5131 . 2 ((𝑋 ∈ V ∧ 𝑅 ∈ V) → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))))
325, 14, 31pm5.21nd 979 1 (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 (𝐹‘(𝑅 “ {𝑦})) = 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  wral 3042  Vcvv 3332  wss 3707  {csn 4313   class class class wbr 4796  {copab 4856   We wwe 5216   × cxp 5256  ccnv 5257  cima 5261  Rel wrel 5263  cfv 6041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fv 6049
This theorem is referenced by:  canth4  9653  canthnumlem  9654  canthp1lem2  9659
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