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Theorem mpomptsx 8105
Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
mpomptsx (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({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 mpomptsx
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3492 . . . . . 6 𝑢 ∈ V
2 vex 3492 . . . . . 6 𝑣 ∈ V
31, 2op1std 8040 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) = 𝑢)
43csbeq1d 3925 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶)
51, 2op2ndd 8041 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) = 𝑣)
65csbeq1d 3925 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
76csbeq2dv 3928 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
84, 7eqtrd 2780 . . 3 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
98mpomptx 7563 . 2 (𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
10 nfcv 2908 . . . 4 𝑢({𝑥} × 𝐵)
11 nfcv 2908 . . . . 5 𝑥{𝑢}
12 nfcsb1v 3946 . . . . 5 𝑥𝑢 / 𝑥𝐵
1311, 12nfxp 5733 . . . 4 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
14 sneq 4658 . . . . 5 (𝑥 = 𝑢 → {𝑥} = {𝑢})
15 csbeq1a 3935 . . . . 5 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
1614, 15xpeq12d 5731 . . . 4 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
1710, 13, 16cbviun 5059 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
1817mpteq1i 5262 . 2 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶) = (𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
19 nfcv 2908 . . 3 𝑢𝐵
20 nfcv 2908 . . 3 𝑢𝐶
21 nfcv 2908 . . 3 𝑣𝐶
22 nfcsb1v 3946 . . 3 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
23 nfcv 2908 . . . 4 𝑦𝑢
24 nfcsb1v 3946 . . . 4 𝑦𝑣 / 𝑦𝐶
2523, 24nfcsbw 3948 . . 3 𝑦𝑢 / 𝑥𝑣 / 𝑦𝐶
26 csbeq1a 3935 . . . 4 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
27 csbeq1a 3935 . . . 4 (𝑥 = 𝑢𝑣 / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2826, 27sylan9eqr 2802 . . 3 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2919, 12, 20, 21, 22, 25, 15, 28cbvmpox 7543 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
309, 18, 293eqtr4ri 2779 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  csb 3921  {csn 4648  cop 4654   ciun 5015  cmpt 5249   × cxp 5698  cfv 6573  cmpo 7450  1st c1st 8028  2nd c2nd 8029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031
This theorem is referenced by:  mpompts  8106  ovmptss  8134
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