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Theorem dfoprab3 5740
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 5739 . 2 {⟨⟨x, y⟩, z⟩ ∣ ψ} = {⟨w, z⟩ ∣ (w (V × V) [(1stw) / x][(2ndw) / y]ψ)}
2 vex 2538 . . . . . 6 w V
3 1stexg 5717 . . . . . 6 (w V → (1stw) V)
42, 3ax-mp 7 . . . . 5 (1stw) V
5 2ndexg 5718 . . . . . 6 (w V → (2ndw) V)
62, 5ax-mp 7 . . . . 5 (2ndw) V
7 eqcom 2024 . . . . . . . . . 10 (x = (1stw) ↔ (1stw) = x)
8 eqcom 2024 . . . . . . . . . 10 (y = (2ndw) ↔ (2ndw) = y)
97, 8anbi12i 436 . . . . . . . . 9 ((x = (1stw) y = (2ndw)) ↔ ((1stw) = x (2ndw) = y))
10 eqopi 5721 . . . . . . . . 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 2807 . . . 4 (w (V × V) → ([(1stw) / x][(2ndw) / y]ψφ))
1716pm5.32i 430 . . 3 ((w (V × V) [(1stw) / x][(2ndw) / y]ψ) ↔ (w (V × V) φ))
1817opabbii 3798 . 2 {⟨w, z⟩ ∣ (w (V × V) [(1stw) / x][(2ndw) / y]ψ)} = {⟨w, z⟩ ∣ (w (V × V) φ)}
191, 18eqtr2i 2043 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 1228   wcel 1374  Vcvv 2535  [wsbc 2741  cop 3353  {copab 3791   × cxp 4270  cfv 4829  {coprab 5437  1st c1st 5688  2nd c2nd 5689
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-oprab 5440  df-1st 5690  df-2nd 5691
This theorem is referenced by:  dfoprab4  5741  df1st2  5763  df2nd2  5764
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