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

Theorem fpwwe2lem3 10569
Description: Lemma for fpwwe2 10579. (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 10568 . . . . 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 4596 . . . . . 6 (𝑦 = 𝐵 → {𝑦} = {𝐵})
87imaeq2d 6013 . . . . 5 (𝑦 = 𝐵 → (𝑅 “ {𝑦}) = (𝑅 “ {𝐵}))
9 eqeq2 2748 . . . . 5 (𝑦 = 𝐵 → ((𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵))
108, 9sbceqbid 3746 . . . 4 (𝑦 = 𝐵 → ([(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦[(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵))
1110rspccva 3580 . . 3 ((∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦𝐵𝑋) → [(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵)
126, 11sylan 580 . 2 ((𝜑𝐵𝑋) → [(𝑅 “ {𝐵}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵)
13 cnvimass 6033 . . . . 5 (𝑅 “ {𝐵}) ⊆ dom 𝑅
142relopabiv 5776 . . . . . . 7 Rel 𝑊
1514brrelex2i 5689 . . . . . 6 (𝑋𝑊𝑅𝑅 ∈ V)
16 dmexg 7840 . . . . . 6 (𝑅 ∈ V → dom 𝑅 ∈ V)
171, 15, 163syl 18 . . . . 5 (𝜑 → dom 𝑅 ∈ V)
18 ssexg 5280 . . . . 5 (((𝑅 “ {𝐵}) ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → (𝑅 “ {𝐵}) ∈ V)
1913, 17, 18sylancr 587 . . . 4 (𝜑 → (𝑅 “ {𝐵}) ∈ V)
20 id 22 . . . . . . 7 (𝑢 = (𝑅 “ {𝐵}) → 𝑢 = (𝑅 “ {𝐵}))
2120sqxpeqd 5665 . . . . . . . 8 (𝑢 = (𝑅 “ {𝐵}) → (𝑢 × 𝑢) = ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))
2221ineq2d 4172 . . . . . . 7 (𝑢 = (𝑅 “ {𝐵}) → (𝑅 ∩ (𝑢 × 𝑢)) = (𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵}))))
2320, 22oveq12d 7375 . . . . . 6 (𝑢 = (𝑅 “ {𝐵}) → (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))))
2423eqeq1d 2738 . . . . 5 (𝑢 = (𝑅 “ {𝐵}) → ((𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝐵 ↔ ((𝑅 “ {𝐵})𝐹(𝑅 ∩ ((𝑅 “ {𝐵}) × (𝑅 “ {𝐵})))) = 𝐵))
2524sbcieg 3779 . . . 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 481 . 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 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  [wsbc 3739  cin 3909  wss 3910  {csn 4586   class class class wbr 5105  {copab 5167   We wwe 5587   × cxp 5631  ccnv 5632  dom cdm 5633  cima 5636  (class class class)co 7357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fv 6504  df-ov 7360
This theorem is referenced by:  fpwwe2lem7  10573  fpwwe2lem11  10577  fpwwe2lem12  10578  fpwwe2  10579  canthwelem  10586  pwfseqlem4  10598
  Copyright terms: Public domain W3C validator