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Theorem dfoprab3s 5960
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 5696 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2 nfsbc1v 2858 . . . . 5 𝑥[(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑
3219.41 1621 . . . 4 (∃𝑥(∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
4 sbcopeq1a 5957 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → ([(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑𝜑))
54pm5.32i 442 . . . . . . 7 ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
65exbii 1541 . . . . . 6 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
7 nfcv 2228 . . . . . . . 8 𝑦(1st𝑤)
8 nfsbc1v 2858 . . . . . . . 8 𝑦[(2nd𝑤) / 𝑦]𝜑
97, 8nfsbc 2860 . . . . . . 7 𝑦[(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑
10919.41 1621 . . . . . 6 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
116, 10bitr3i 184 . . . . 5 (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
1211exbii 1541 . . . 4 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
13 elvv 4500 . . . . 5 (𝑤 ∈ (V × V) ↔ ∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
1413anbi1i 446 . . . 4 ((𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑) ↔ (∃𝑥𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
153, 12, 143bitr4i 210 . . 3 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑))
1615opabbii 3905 . 2 {⟨𝑤, 𝑧⟩ ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑)}
171, 16eqtri 2108 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜑)}
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  [wsbc 2840  cop 3449  {copab 3898   × cxp 4436  cfv 5015  {coprab 5653  1st c1st 5909  2nd c2nd 5910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-oprab 5656  df-1st 5911  df-2nd 5912
This theorem is referenced by:  dfoprab3  5961
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