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Theorem dfoprab3 5817
 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 5816 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓)}
2 vex 2560 . . . . . 6 𝑤 ∈ V
3 1stexg 5794 . . . . . 6 (𝑤 ∈ V → (1st𝑤) ∈ V)
42, 3ax-mp 7 . . . . 5 (1st𝑤) ∈ V
5 2ndexg 5795 . . . . . 6 (𝑤 ∈ V → (2nd𝑤) ∈ V)
62, 5ax-mp 7 . . . . 5 (2nd𝑤) ∈ V
7 eqcom 2042 . . . . . . . . . 10 (𝑥 = (1st𝑤) ↔ (1st𝑤) = 𝑥)
8 eqcom 2042 . . . . . . . . . 10 (𝑦 = (2nd𝑤) ↔ (2nd𝑤) = 𝑦)
97, 8anbi12i 433 . . . . . . . . 9 ((𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤)) ↔ ((1st𝑤) = 𝑥 ∧ (2nd𝑤) = 𝑦))
10 eqopi 5798 . . . . . . . . 9 ((𝑤 ∈ (V × V) ∧ ((1st𝑤) = 𝑥 ∧ (2nd𝑤) = 𝑦)) → 𝑤 = ⟨𝑥, 𝑦⟩)
119, 10sylan2b 271 . . . . . . . 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 129 . . . . . 6 ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤))) → (𝜓𝜑))
1514ex 108 . . . . 5 (𝑤 ∈ (V × V) → ((𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤)) → (𝜓𝜑)))
164, 6, 15sbc2iedv 2830 . . . 4 (𝑤 ∈ (V × V) → ([(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓𝜑))
1716pm5.32i 427 . . 3 ((𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓) ↔ (𝑤 ∈ (V × V) ∧ 𝜑))
1817opabbii 3824 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)}
191, 18eqtr2i 2061 1 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  Vcvv 2557  [wsbc 2764  ⟨cop 3378  {copab 3817   × cxp 4343  ‘cfv 4902  {coprab 5513  1st c1st 5765  2nd c2nd 5766 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-oprab 5516  df-1st 5767  df-2nd 5768 This theorem is referenced by:  dfoprab4  5818  df1st2  5840  df2nd2  5841
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