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Theorem dfoprab3s 7267
Description: A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dfoprab3s {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑)}
Distinct variable groups:   𝜑,𝑤   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem dfoprab3s
StepHypRef Expression
1 dfoprab2 6743 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfsbc1v 3488 . . . . 5 𝑥[(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑
3219.41 2141 . . . 4 (∃𝑥(∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
4 sbcopeq1a 7264 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → ([(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑𝜑))
54pm5.32i 670 . . . . . . 7 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
65exbii 1814 . . . . . 6 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
7 nfcv 2793 . . . . . . . 8 𝑦(1st𝑤)
8 nfsbc1v 3488 . . . . . . . 8 𝑦[(2nd𝑤) / 𝑦]𝜑
97, 8nfsbc 3490 . . . . . . 7 𝑦[(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑
10919.41 2141 . . . . . 6 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
116, 10bitr3i 266 . . . . 5 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
1211exbii 1814 . . . 4 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
13 elvv 5211 . . . . 5 (𝑤 ∈ (V × V) ↔ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
1413anbi1i 731 . . . 4 ((𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
153, 12, 143bitr4i 292 . . 3 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
1615opabbii 4750 . 2 {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑)}
171, 16eqtri 2673 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑)}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  [wsbc 3468  cop 4216  {copab 4745   × cxp 5141  cfv 5926  {coprab 6691  1st c1st 7208  2nd c2nd 7209
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-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-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-oprab 6694  df-1st 7210  df-2nd 7211
This theorem is referenced by:  dfoprab3  7268
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