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Theorem mpomptsx 8089
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 3484 . . . . . 6 𝑢 ∈ V
2 vex 3484 . . . . . 6 𝑣 ∈ V
31, 2op1std 8024 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) = 𝑢)
43csbeq1d 3903 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶)
51, 2op2ndd 8025 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) = 𝑣)
65csbeq1d 3903 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
76csbeq2dv 3906 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
84, 7eqtrd 2777 . . 3 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
98mpomptx 7546 . 2 (𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
10 nfcv 2905 . . . 4 𝑢({𝑥} × 𝐵)
11 nfcv 2905 . . . . 5 𝑥{𝑢}
12 nfcsb1v 3923 . . . . 5 𝑥𝑢 / 𝑥𝐵
1311, 12nfxp 5718 . . . 4 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
14 sneq 4636 . . . . 5 (𝑥 = 𝑢 → {𝑥} = {𝑢})
15 csbeq1a 3913 . . . . 5 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
1614, 15xpeq12d 5716 . . . 4 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
1710, 13, 16cbviun 5036 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
1817mpteq1i 5238 . 2 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶) = (𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
19 nfcv 2905 . . 3 𝑢𝐵
20 nfcv 2905 . . 3 𝑢𝐶
21 nfcv 2905 . . 3 𝑣𝐶
22 nfcsb1v 3923 . . 3 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
23 nfcv 2905 . . . 4 𝑦𝑢
24 nfcsb1v 3923 . . . 4 𝑦𝑣 / 𝑦𝐶
2523, 24nfcsbw 3925 . . 3 𝑦𝑢 / 𝑥𝑣 / 𝑦𝐶
26 csbeq1a 3913 . . . 4 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
27 csbeq1a 3913 . . . 4 (𝑥 = 𝑢𝑣 / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2826, 27sylan9eqr 2799 . . 3 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2919, 12, 20, 21, 22, 25, 15, 28cbvmpox 7526 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
309, 18, 293eqtr4ri 2776 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  csb 3899  {csn 4626  cop 4632   ciun 4991  cmpt 5225   × cxp 5683  cfv 6561  cmpo 7433  1st c1st 8012  2nd c2nd 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015
This theorem is referenced by:  mpompts  8090  ovmptss  8118
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