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Theorem fpwwe2lem3 9493
Description: Lemma for fpwwe2 9503. (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 9492 . . . . 5 (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
51, 4mpbid 222 . . . 4 (𝜑 → ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
65simprrd 812 . . 3 (𝜑 → ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)
7 sneq 4220 . . . . . 6 (𝑦 = 𝐵 → {𝑦} = {𝐵})
87imaeq2d 5501 . . . . 5 (𝑦 = 𝐵 → (𝑅 “ {𝑦}) = (𝑅 “ {𝐵}))
9 eqeq2 2662 . . . . 5 (𝑦 = 𝐵 → ((𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵))
108, 9sbceqbid 3475 . . . 4 (𝑦 = 𝐵 → ([(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦[(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵))
1110rspccva 3339 . . 3 ((∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦𝐵𝑋) → [(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵)
126, 11sylan 487 . 2 ((𝜑𝐵𝑋) → [(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵)
13 cnvimass 5520 . . . . 5 (𝑅 “ {𝐵}) ⊆ dom 𝑅
142relopabi 5278 . . . . . . 7 Rel 𝑊
1514brrelex2i 5193 . . . . . 6 (𝑋𝑊𝑅𝑅 ∈ V)
16 dmexg 7139 . . . . . 6 (𝑅 ∈ V → dom 𝑅 ∈ V)
171, 15, 163syl 18 . . . . 5 (𝜑 → dom 𝑅 ∈ V)
18 ssexg 4837 . . . . 5 (((𝑅 “ {𝐵}) ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → (𝑅 “ {𝐵}) ∈ V)
1913, 17, 18sylancr 696 . . . 4 (𝜑 → (𝑅 “ {𝐵}) ∈ V)
20 id 22 . . . . . . 7 (𝑢 = (𝑅 “ {𝐵}) → 𝑢 = (𝑅 “ {𝐵}))
2120sqxpeqd 5175 . . . . . . . 8 (𝑢 = (𝑅 “ {𝐵}) → (𝑢 × 𝑢) = ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))
2221ineq2d 3847 . . . . . . 7 (𝑢 = (𝑅 “ {𝐵}) → (𝑅 ∩ (𝑢 × 𝑢)) = (𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵}))))
2320, 22oveq12d 6708 . . . . . 6 (𝑢 = (𝑅 “ {𝐵}) → (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))))
2423eqeq1d 2653 . . . . 5 (𝑢 = (𝑅 “ {𝐵}) → ((𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵))
2524sbcieg 3501 . . . 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 480 . 2 ((𝜑𝐵𝑋) → ([(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵))
2812, 27mpbid 222 1 ((𝜑𝐵𝑋) → ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  [wsbc 3468  cin 3606  wss 3607  {csn 4210   class class class wbr 4685  {copab 4745   We wwe 5101   × cxp 5141  ccnv 5142  dom cdm 5143  cima 5146  (class class class)co 6690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fv 5934  df-ov 6693
This theorem is referenced by:  fpwwe2lem8  9497  fpwwe2lem12  9501  fpwwe2lem13  9502  fpwwe2  9503  canthwelem  9510  pwfseqlem4  9522
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