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Theorem dfoprab3s 6088
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 5818 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfsbc1v 2927 . . . . 5 𝑥[(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑
3219.41 1664 . . . 4 (∃𝑥(∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
4 sbcopeq1a 6085 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → ([(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑𝜑))
54pm5.32i 449 . . . . . . 7 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
65exbii 1584 . . . . . 6 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
7 nfcv 2281 . . . . . . . 8 𝑦(1st𝑤)
8 nfsbc1v 2927 . . . . . . . 8 𝑦[(2nd𝑤) / 𝑦]𝜑
97, 8nfsbc 2929 . . . . . . 7 𝑦[(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑
10919.41 1664 . . . . . 6 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
116, 10bitr3i 185 . . . . 5 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
1211exbii 1584 . . . 4 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
13 elvv 4601 . . . . 5 (𝑤 ∈ (V × V) ↔ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
1413anbi1i 453 . . . 4 ((𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
153, 12, 143bitr4i 211 . . 3 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
1615opabbii 3995 . 2 {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑)}
171, 16eqtri 2160 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑)}
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1331  wex 1468  wcel 1480  Vcvv 2686  [wsbc 2909  cop 3530  {copab 3988   × cxp 4537  cfv 5123  {coprab 5775  1st c1st 6036  2nd c2nd 6037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-oprab 5778  df-1st 6038  df-2nd 6039
This theorem is referenced by:  dfoprab3  6089
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