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Theorem dfoprab3s 8055
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 7475 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfsbc1v 3788 . . . . 5 𝑥[(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑
3219.41 2223 . . . 4 (∃𝑥(∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
4 sbcopeq1a 8051 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → ([(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑𝜑))
54pm5.32i 573 . . . . . . 7 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
65exbii 1842 . . . . . 6 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
7 nfcv 2892 . . . . . . . 8 𝑦(1st𝑤)
8 nfsbc1v 3788 . . . . . . . 8 𝑦[(2nd𝑤) / 𝑦]𝜑
97, 8nfsbcw 3790 . . . . . . 7 𝑦[(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑
10919.41 2223 . . . . . 6 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
116, 10bitr3i 276 . . . . 5 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
1211exbii 1842 . . . 4 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
13 elvv 5746 . . . . 5 (𝑤 ∈ (V × V) ↔ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
1413anbi1i 622 . . . 4 ((𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
153, 12, 143bitr4i 302 . . 3 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
1615opabbii 5210 . 2 {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑)}
171, 16eqtri 2753 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑)}
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wex 1773  wcel 2098  Vcvv 3463  [wsbc 3768  cop 4630  {copab 5205   × cxp 5670  cfv 6543  {coprab 7417  1st c1st 7989  2nd c2nd 7990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-oprab 7420  df-1st 7991  df-2nd 7992
This theorem is referenced by:  dfoprab3  8056
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