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Theorem mpt2mptsx 5765
Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
mpt2mptsx (x A, y B𝐶) = (z x A ({x} × B) ↦ (1stz) / x(2ndz) / y𝐶)
Distinct variable groups:   x,y,z,A   y,B,z   z,𝐶
Allowed substitution hints:   B(x)   𝐶(x,y)

Proof of Theorem mpt2mptsx
Dummy variables v u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . 6 u V
2 vex 2554 . . . . . 6 v V
31, 2op1std 5717 . . . . 5 (z = ⟨u, v⟩ → (1stz) = u)
43csbeq1d 2852 . . . 4 (z = ⟨u, v⟩ → (1stz) / x(2ndz) / y𝐶 = u / x(2ndz) / y𝐶)
51, 2op2ndd 5718 . . . . . 6 (z = ⟨u, v⟩ → (2ndz) = v)
65csbeq1d 2852 . . . . 5 (z = ⟨u, v⟩ → (2ndz) / y𝐶 = v / y𝐶)
76csbeq2dv 2869 . . . 4 (z = ⟨u, v⟩ → u / x(2ndz) / y𝐶 = u / xv / y𝐶)
84, 7eqtrd 2069 . . 3 (z = ⟨u, v⟩ → (1stz) / x(2ndz) / y𝐶 = u / xv / y𝐶)
98mpt2mptx 5537 . 2 (z u A ({u} × u / xB) ↦ (1stz) / x(2ndz) / y𝐶) = (u A, v u / xBu / xv / y𝐶)
10 nfcv 2175 . . . 4 u({x} × B)
11 nfcv 2175 . . . . 5 x{u}
12 nfcsb1v 2876 . . . . 5 xu / xB
1311, 12nfxp 4314 . . . 4 x({u} × u / xB)
14 sneq 3378 . . . . 5 (x = u → {x} = {u})
15 csbeq1a 2854 . . . . 5 (x = uB = u / xB)
1614, 15xpeq12d 4313 . . . 4 (x = u → ({x} × B) = ({u} × u / xB))
1710, 13, 16cbviun 3685 . . 3 x A ({x} × B) = u A ({u} × u / xB)
18 mpteq1 3832 . . 3 ( x A ({x} × B) = u A ({u} × u / xB) → (z x A ({x} × B) ↦ (1stz) / x(2ndz) / y𝐶) = (z u A ({u} × u / xB) ↦ (1stz) / x(2ndz) / y𝐶))
1917, 18ax-mp 7 . 2 (z x A ({x} × B) ↦ (1stz) / x(2ndz) / y𝐶) = (z u A ({u} × u / xB) ↦ (1stz) / x(2ndz) / y𝐶)
20 nfcv 2175 . . 3 uB
21 nfcv 2175 . . 3 u𝐶
22 nfcv 2175 . . 3 v𝐶
23 nfcsb1v 2876 . . 3 xu / xv / y𝐶
24 nfcv 2175 . . . 4 yu
25 nfcsb1v 2876 . . . 4 yv / y𝐶
2624, 25nfcsb 2878 . . 3 yu / xv / y𝐶
27 csbeq1a 2854 . . . 4 (y = v𝐶 = v / y𝐶)
28 csbeq1a 2854 . . . 4 (x = uv / y𝐶 = u / xv / y𝐶)
2927, 28sylan9eqr 2091 . . 3 ((x = u y = v) → 𝐶 = u / xv / y𝐶)
3020, 12, 21, 22, 23, 26, 15, 29cbvmpt2x 5524 . 2 (x A, y B𝐶) = (u A, v u / xBu / xv / y𝐶)
319, 19, 303eqtr4ri 2068 1 (x A, y B𝐶) = (z x A ({x} × B) ↦ (1stz) / x(2ndz) / y𝐶)
Colors of variables: wff set class
Syntax hints:   = wceq 1242  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-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-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710
This theorem is referenced by:  mpt2mpts  5766  mpt2fvex  5771
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