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Theorem fpwwe2lem3 9859
Description: Lemma for fpwwe2 9869. (Contributed by Mario Carneiro, 19-May-2015.)
Hypotheses
Ref Expression
fpwwe2.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2 (𝜑𝐴 ∈ V)
fpwwe2lem4.4 (𝜑𝑋𝑊𝑅)
Assertion
Ref Expression
fpwwe2lem3 ((𝜑𝐵𝑋) → ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵)
Distinct variable groups:   𝑦,𝑢,𝐵   𝑢,𝑟,𝑥,𝑦,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝜑,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑥   𝑅,𝑟,𝑢,𝑥,𝑦   𝑊,𝑟,𝑢,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑦,𝑢)   𝐵(𝑥,𝑟)

Proof of Theorem fpwwe2lem3
StepHypRef Expression
1 fpwwe2lem4.4 . . . . 5 (𝜑𝑋𝑊𝑅)
2 fpwwe2.1 . . . . . 6 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
3 fpwwe2.2 . . . . . 6 (𝜑𝐴 ∈ V)
42, 3fpwwe2lem2 9858 . . . . 5 (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
51, 4mpbid 224 . . . 4 (𝜑 → ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
65simprrd 762 . . 3 (𝜑 → ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)
7 sneq 4454 . . . . . 6 (𝑦 = 𝐵 → {𝑦} = {𝐵})
87imaeq2d 5775 . . . . 5 (𝑦 = 𝐵 → (𝑅 “ {𝑦}) = (𝑅 “ {𝐵}))
9 eqeq2 2791 . . . . 5 (𝑦 = 𝐵 → ((𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵))
108, 9sbceqbid 3690 . . . 4 (𝑦 = 𝐵 → ([(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦[(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵))
1110rspccva 3536 . . 3 ((∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦𝐵𝑋) → [(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵)
126, 11sylan 572 . 2 ((𝜑𝐵𝑋) → [(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵)
13 cnvimass 5794 . . . . 5 (𝑅 “ {𝐵}) ⊆ dom 𝑅
142relopabi 5548 . . . . . . 7 Rel 𝑊
1514brrelex2i 5463 . . . . . 6 (𝑋𝑊𝑅𝑅 ∈ V)
16 dmexg 7434 . . . . . 6 (𝑅 ∈ V → dom 𝑅 ∈ V)
171, 15, 163syl 18 . . . . 5 (𝜑 → dom 𝑅 ∈ V)
18 ssexg 5087 . . . . 5 (((𝑅 “ {𝐵}) ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → (𝑅 “ {𝐵}) ∈ V)
1913, 17, 18sylancr 579 . . . 4 (𝜑 → (𝑅 “ {𝐵}) ∈ V)
20 id 22 . . . . . . 7 (𝑢 = (𝑅 “ {𝐵}) → 𝑢 = (𝑅 “ {𝐵}))
2120sqxpeqd 5443 . . . . . . . 8 (𝑢 = (𝑅 “ {𝐵}) → (𝑢 × 𝑢) = ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))
2221ineq2d 4079 . . . . . . 7 (𝑢 = (𝑅 “ {𝐵}) → (𝑅 ∩ (𝑢 × 𝑢)) = (𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵}))))
2320, 22oveq12d 7000 . . . . . 6 (𝑢 = (𝑅 “ {𝐵}) → (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))))
2423eqeq1d 2782 . . . . 5 (𝑢 = (𝑅 “ {𝐵}) → ((𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵))
2524sbcieg 3716 . . . 4 ((𝑅 “ {𝐵}) ∈ V → ([(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵))
2619, 25syl 17 . . 3 (𝜑 → ([(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵))
2726adantr 473 . 2 ((𝜑𝐵𝑋) → ([(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵))
2812, 27mpbid 224 1 ((𝜑𝐵𝑋) → ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1508  wcel 2051  wral 3090  Vcvv 3417  [wsbc 3683  cin 3830  wss 3831  {csn 4444   class class class wbr 4934  {copab 4996   We wwe 5369   × cxp 5409  ccnv 5410  dom cdm 5411  cima 5414  (class class class)co 6982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3419  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-op 4451  df-uni 4718  df-br 4935  df-opab 4997  df-po 5330  df-so 5331  df-fr 5370  df-we 5372  df-xp 5417  df-rel 5418  df-cnv 5419  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-iota 6157  df-fv 6201  df-ov 6985
This theorem is referenced by:  fpwwe2lem8  9863  fpwwe2lem12  9867  fpwwe2lem13  9868  fpwwe2  9869  canthwelem  9876  pwfseqlem4  9888
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