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Theorem mpt2mpt 5538
Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
mpt2mpt.1 (z = ⟨x, y⟩ → 𝐶 = 𝐷)
Assertion
Ref Expression
mpt2mpt (z (A × B) ↦ 𝐶) = (x A, y B𝐷)
Distinct variable groups:   x,y,z,A   y,B,z   x,𝐶,y   z,𝐷   x,B
Allowed substitution hints:   𝐶(z)   𝐷(x,y)

Proof of Theorem mpt2mpt
StepHypRef Expression
1 iunxpconst 4343 . . 3 x A ({x} × B) = (A × B)
2 mpteq1 3832 . . 3 ( x A ({x} × B) = (A × B) → (z x A ({x} × B) ↦ 𝐶) = (z (A × B) ↦ 𝐶))
31, 2ax-mp 7 . 2 (z x A ({x} × B) ↦ 𝐶) = (z (A × B) ↦ 𝐶)
4 mpt2mpt.1 . . 3 (z = ⟨x, y⟩ → 𝐶 = 𝐷)
54mpt2mptx 5537 . 2 (z x A ({x} × B) ↦ 𝐶) = (x A, y B𝐷)
63, 5eqtr3i 2059 1 (z (A × B) ↦ 𝐶) = (x A, y B𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  {csn 3367  cop 3370   ciun 3648  cmpt 3809   × cxp 4286  cmpt2 5457
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-bndl 1396  ax-4 1397  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
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  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-iun 3650  df-opab 3810  df-mpt 3811  df-xp 4294  df-rel 4295  df-oprab 5459  df-mpt2 5460
This theorem is referenced by:  fnovim  5551  fmpt2co  5779
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