MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fpwwe2lem3 Structured version   Visualization version   GIF version

Theorem fpwwe2lem3 10676
Description: Lemma for fpwwe2 10686. (Contributed by Mario Carneiro, 19-May-2015.) (Revised by AV, 20-Jul-2024.)
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 (𝜑𝐴𝑉)
fpwwe2lem3.4 (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)   𝑉(𝑥,𝑦,𝑢,𝑟)

Proof of Theorem fpwwe2lem3
StepHypRef Expression
1 fpwwe2lem3.4 . . . . 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 (𝜑𝐴𝑉)
42, 3fpwwe2lem2 10675 . . . . 5 (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
51, 4mpbid 231 . . . 4 (𝜑 → ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
65simprrd 772 . . 3 (𝜑 → ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)
7 sneq 4643 . . . . . 6 (𝑦 = 𝐵 → {𝑦} = {𝐵})
87imaeq2d 6069 . . . . 5 (𝑦 = 𝐵 → (𝑅 “ {𝑦}) = (𝑅 “ {𝐵}))
9 eqeq2 2738 . . . . 5 (𝑦 = 𝐵 → ((𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵))
108, 9sbceqbid 3783 . . . 4 (𝑦 = 𝐵 → ([(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦[(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵))
1110rspccva 3607 . . 3 ((∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦𝐵𝑋) → [(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵)
126, 11sylan 578 . 2 ((𝜑𝐵𝑋) → [(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵)
13 cnvimass 6091 . . . . 5 (𝑅 “ {𝐵}) ⊆ dom 𝑅
142relopabiv 5826 . . . . . . 7 Rel 𝑊
1514brrelex2i 5739 . . . . . 6 (𝑋𝑊𝑅𝑅 ∈ V)
16 dmexg 7914 . . . . . 6 (𝑅 ∈ V → dom 𝑅 ∈ V)
171, 15, 163syl 18 . . . . 5 (𝜑 → dom 𝑅 ∈ V)
18 ssexg 5328 . . . . 5 (((𝑅 “ {𝐵}) ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → (𝑅 “ {𝐵}) ∈ V)
1913, 17, 18sylancr 585 . . . 4 (𝜑 → (𝑅 “ {𝐵}) ∈ V)
20 id 22 . . . . . . 7 (𝑢 = (𝑅 “ {𝐵}) → 𝑢 = (𝑅 “ {𝐵}))
2120sqxpeqd 5714 . . . . . . . 8 (𝑢 = (𝑅 “ {𝐵}) → (𝑢 × 𝑢) = ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))
2221ineq2d 4213 . . . . . . 7 (𝑢 = (𝑅 “ {𝐵}) → (𝑅 ∩ (𝑢 × 𝑢)) = (𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵}))))
2320, 22oveq12d 7442 . . . . . 6 (𝑢 = (𝑅 “ {𝐵}) → (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))))
2423eqeq1d 2728 . . . . 5 (𝑢 = (𝑅 “ {𝐵}) → ((𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵))
2524sbcieg 3817 . . . 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 479 . 2 ((𝜑𝐵𝑋) → ([(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵))
2812, 27mpbid 231 1 ((𝜑𝐵𝑋) → ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  Vcvv 3462  [wsbc 3776  cin 3946  wss 3947  {csn 4633   class class class wbr 5153  {copab 5215   We wwe 5636   × cxp 5680  ccnv 5681  dom cdm 5682  cima 5685  (class class class)co 7424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fv 6562  df-ov 7427
This theorem is referenced by:  fpwwe2lem7  10680  fpwwe2lem11  10684  fpwwe2lem12  10685  fpwwe2  10686  canthwelem  10693  pwfseqlem4  10705
  Copyright terms: Public domain W3C validator