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Theorem dfmpt2 5767
Description: Alternate definition for the "maps to" notation df-mpt2 5441 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfmpt2.1 𝐶 V
Assertion
Ref Expression
dfmpt2 (x A, y B𝐶) = x A y B {⟨⟨x, y⟩, 𝐶⟩}
Distinct variable groups:   x,y,A   x,B,y
Allowed substitution hints:   𝐶(x,y)

Proof of Theorem dfmpt2
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 mpt2mpts 5747 . 2 (x A, y B𝐶) = (w (A × B) ↦ (1stw) / x(2ndw) / y𝐶)
2 vex 2538 . . . . 5 w V
3 1stexg 5717 . . . . 5 (w V → (1stw) V)
42, 3ax-mp 7 . . . 4 (1stw) V
5 2ndexg 5718 . . . . . 6 (w V → (2ndw) V)
62, 5ax-mp 7 . . . . 5 (2ndw) V
7 dfmpt2.1 . . . . 5 𝐶 V
86, 7csbexOLD 3860 . . . 4 (2ndw) / y𝐶 V
94, 8csbexOLD 3860 . . 3 (1stw) / x(2ndw) / y𝐶 V
109dfmpt 5265 . 2 (w (A × B) ↦ (1stw) / x(2ndw) / y𝐶) = w (A × B){⟨w, (1stw) / x(2ndw) / y𝐶⟩}
11 nfcv 2160 . . . . 5 xw
12 nfcsb1v 2859 . . . . 5 x(1stw) / x(2ndw) / y𝐶
1311, 12nfop 3539 . . . 4 xw, (1stw) / x(2ndw) / y𝐶
1413nfsn 3404 . . 3 x{⟨w, (1stw) / x(2ndw) / y𝐶⟩}
15 nfcv 2160 . . . . 5 yw
16 nfcv 2160 . . . . . 6 y(1stw)
17 nfcsb1v 2859 . . . . . 6 y(2ndw) / y𝐶
1816, 17nfcsb 2861 . . . . 5 y(1stw) / x(2ndw) / y𝐶
1915, 18nfop 3539 . . . 4 yw, (1stw) / x(2ndw) / y𝐶
2019nfsn 3404 . . 3 y{⟨w, (1stw) / x(2ndw) / y𝐶⟩}
21 nfcv 2160 . . 3 w{⟨⟨x, y⟩, 𝐶⟩}
22 id 19 . . . . 5 (w = ⟨x, y⟩ → w = ⟨x, y⟩)
23 csbopeq1a 5737 . . . . 5 (w = ⟨x, y⟩ → (1stw) / x(2ndw) / y𝐶 = 𝐶)
2422, 23opeq12d 3531 . . . 4 (w = ⟨x, y⟩ → ⟨w, (1stw) / x(2ndw) / y𝐶⟩ = ⟨⟨x, y⟩, 𝐶⟩)
2524sneqd 3363 . . 3 (w = ⟨x, y⟩ → {⟨w, (1stw) / x(2ndw) / y𝐶⟩} = {⟨⟨x, y⟩, 𝐶⟩})
2614, 20, 21, 25iunxpf 4411 . 2 w (A × B){⟨w, (1stw) / x(2ndw) / y𝐶⟩} = x A y B {⟨⟨x, y⟩, 𝐶⟩}
271, 10, 263eqtri 2046 1 (x A, y B𝐶) = x A y B {⟨⟨x, y⟩, 𝐶⟩}
Colors of variables: wff set class
Syntax hints:   = wceq 1228   wcel 1374  Vcvv 2535  csb 2829  {csn 3350  cop 3353   ciun 3631  cmpt 3792   × cxp 4270  cfv 4829  cmpt2 5438  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-reu 2291  df-v 2537  df-sbc 2742  df-csb 2830  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-iun 3633  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-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691
This theorem is referenced by: (None)
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