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Theorem dfoprab3 5844
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 5843 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓)}
2 vex 2577 . . . . . 6 𝑤 ∈ V
3 1stexg 5821 . . . . . 6 (𝑤 ∈ V → (1st𝑤) ∈ V)
42, 3ax-mp 7 . . . . 5 (1st𝑤) ∈ V
5 2ndexg 5822 . . . . . 6 (𝑤 ∈ V → (2nd𝑤) ∈ V)
62, 5ax-mp 7 . . . . 5 (2nd𝑤) ∈ V
7 eqcom 2058 . . . . . . . . . 10 (𝑥 = (1st𝑤) ↔ (1st𝑤) = 𝑥)
8 eqcom 2058 . . . . . . . . . 10 (𝑦 = (2nd𝑤) ↔ (2nd𝑤) = 𝑦)
97, 8anbi12i 441 . . . . . . . . 9 ((𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤)) ↔ ((1st𝑤) = 𝑥 ∧ (2nd𝑤) = 𝑦))
10 eqopi 5825 . . . . . . . . 9 ((𝑤 ∈ (V × V) ∧ ((1st𝑤) = 𝑥 ∧ (2nd𝑤) = 𝑦)) → 𝑤 = ⟨𝑥, 𝑦⟩)
119, 10sylan2b 275 . . . . . . . 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 133 . . . . . 6 ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤))) → (𝜓𝜑))
1514ex 112 . . . . 5 (𝑤 ∈ (V × V) → ((𝑥 = (1st𝑤) ∧ 𝑦 = (2nd𝑤)) → (𝜓𝜑)))
164, 6, 15sbc2iedv 2857 . . . 4 (𝑤 ∈ (V × V) → ([(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓𝜑))
1716pm5.32i 435 . . 3 ((𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓) ↔ (𝑤 ∈ (V × V) ∧ 𝜑))
1817opabbii 3851 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st𝑤) / 𝑥][(2nd𝑤) / 𝑦]𝜓)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)}
191, 18eqtr2i 2077 1 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  Vcvv 2574  [wsbc 2786  cop 3405  {copab 3844   × cxp 4370  cfv 4929  {coprab 5540  1st c1st 5792  2nd c2nd 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fo 4935  df-fv 4937  df-oprab 5543  df-1st 5794  df-2nd 5795
This theorem is referenced by:  dfoprab4  5845  df1st2  5867  df2nd2  5868
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