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Theorem dfoprab3 6082
Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)
Hypothesis
Ref Expression
dfoprab3.1 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
Assertion
Ref Expression
dfoprab3 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Distinct variable groups:   𝑥,𝑦,𝜑   𝜓,𝑤   𝑥,𝑧,𝑤,𝑦
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem dfoprab3
StepHypRef Expression
1 dfoprab3s 6081 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓)}
2 vex 2684 . . . . . 6 𝑤 ∈ V
3 1stexg 6058 . . . . . 6 (𝑤 ∈ V → (1st𝑤) ∈ V)
42, 3ax-mp 5 . . . . 5 (1st𝑤) ∈ V
5 2ndexg 6059 . . . . . 6 (𝑤 ∈ V → (2nd𝑤) ∈ V)
62, 5ax-mp 5 . . . . 5 (2nd𝑤) ∈ V
7 eqcom 2139 . . . . . . . . . 10 (𝑥 = (1st𝑤) ↔ (1st𝑤) = 𝑥)
8 eqcom 2139 . . . . . . . . . 10 (𝑦 = (2nd𝑤) ↔ (2nd𝑤) = 𝑦)
97, 8anbi12i 455 . . . . . . . . 9 ((𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤)) ↔ ((1st𝑤) = 𝑥 ∧ (2nd𝑤) = 𝑦))
10 eqopi 6063 . . . . . . . . 9 ((𝑤 ∈ (V × V) ∧ ((1st𝑤) = 𝑥 ∧ (2nd𝑤) = 𝑦)) → 𝑤 = ⟨𝑥, 𝑦⟩)
119, 10sylan2b 285 . . . . . . . 8 ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤))) → 𝑤 = ⟨𝑥, 𝑦⟩)
12 dfoprab3.1 . . . . . . . 8 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
1311, 12syl 14 . . . . . . 7 ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤))) → (𝜑𝜓))
1413bicomd 140 . . . . . 6 ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤))) → (𝜓𝜑))
1514ex 114 . . . . 5 (𝑤 ∈ (V × V) → ((𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤)) → (𝜓𝜑)))
164, 6, 15sbc2iedv 2976 . . . 4 (𝑤 ∈ (V × V) → ([(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓𝜑))
1716pm5.32i 449 . . 3 ((𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓) ↔ (𝑤 ∈ (V × V) ∧ 𝜑))
1817opabbii 3990 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)}
191, 18eqtr2i 2159 1 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  Vcvv 2681  [wsbc 2904  cop 3525  {copab 3983   × cxp 4532  cfv 5118  {coprab 5768  1st c1st 6029  2nd c2nd 6030
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 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-oprab 5771  df-1st 6031  df-2nd 6032
This theorem is referenced by:  dfoprab4  6083  df1st2  6109  df2nd2  6110
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