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Theorem dfmpt2 5786
Description: Alternate definition for the "maps to" notation df-mpt2 5460 (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 5766 . 2 (x A, y B𝐶) = (w (A × B) ↦ (1stw) / x(2ndw) / y𝐶)
2 vex 2554 . . . . 5 w V
3 1stexg 5736 . . . . 5 (w V → (1stw) V)
42, 3ax-mp 7 . . . 4 (1stw) V
5 2ndexg 5737 . . . . . 6 (w V → (2ndw) V)
62, 5ax-mp 7 . . . . 5 (2ndw) V
7 dfmpt2.1 . . . . 5 𝐶 V
86, 7csbexa 3877 . . . 4 (2ndw) / y𝐶 V
94, 8csbexa 3877 . . 3 (1stw) / x(2ndw) / y𝐶 V
109dfmpt 5283 . 2 (w (A × B) ↦ (1stw) / x(2ndw) / y𝐶) = w (A × B){⟨w, (1stw) / x(2ndw) / y𝐶⟩}
11 nfcv 2175 . . . . 5 xw
12 nfcsb1v 2876 . . . . 5 x(1stw) / x(2ndw) / y𝐶
1311, 12nfop 3556 . . . 4 xw, (1stw) / x(2ndw) / y𝐶
1413nfsn 3421 . . 3 x{⟨w, (1stw) / x(2ndw) / y𝐶⟩}
15 nfcv 2175 . . . . 5 yw
16 nfcv 2175 . . . . . 6 y(1stw)
17 nfcsb1v 2876 . . . . . 6 y(2ndw) / y𝐶
1816, 17nfcsb 2878 . . . . 5 y(1stw) / x(2ndw) / y𝐶
1915, 18nfop 3556 . . . 4 yw, (1stw) / x(2ndw) / y𝐶
2019nfsn 3421 . . 3 y{⟨w, (1stw) / x(2ndw) / y𝐶⟩}
21 nfcv 2175 . . 3 w{⟨⟨x, y⟩, 𝐶⟩}
22 id 19 . . . . 5 (w = ⟨x, y⟩ → w = ⟨x, y⟩)
23 csbopeq1a 5756 . . . . 5 (w = ⟨x, y⟩ → (1stw) / x(2ndw) / y𝐶 = 𝐶)
2422, 23opeq12d 3548 . . . 4 (w = ⟨x, y⟩ → ⟨w, (1stw) / x(2ndw) / y𝐶⟩ = ⟨⟨x, y⟩, 𝐶⟩)
2524sneqd 3380 . . 3 (w = ⟨x, y⟩ → {⟨w, (1stw) / x(2ndw) / y𝐶⟩} = {⟨⟨x, y⟩, 𝐶⟩})
2614, 20, 21, 25iunxpf 4427 . 2 w (A × B){⟨w, (1stw) / x(2ndw) / y𝐶⟩} = x A y B {⟨⟨x, y⟩, 𝐶⟩}
271, 10, 263eqtri 2061 1 (x A, y B𝐶) = x A y B {⟨⟨x, y⟩, 𝐶⟩}
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  Vcvv 2551  csb 2846  {csn 3367  cop 3370   ciun 3648  cmpt 3809   × cxp 4286  cfv 4845  cmpt2 5457  1st c1st 5707  2nd c2nd 5708
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
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