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Theorem mpomptsx 7747
 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 3444 . . . . . 6 𝑢 ∈ V
2 vex 3444 . . . . . 6 𝑣 ∈ V
31, 2op1std 7684 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) = 𝑢)
43csbeq1d 3832 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶)
51, 2op2ndd 7685 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) = 𝑣)
65csbeq1d 3832 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
76csbeq2dv 3835 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
84, 7eqtrd 2833 . . 3 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
98mpomptx 7245 . 2 (𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
10 nfcv 2955 . . . 4 𝑢({𝑥} × 𝐵)
11 nfcv 2955 . . . . 5 𝑥{𝑢}
12 nfcsb1v 3852 . . . . 5 𝑥𝑢 / 𝑥𝐵
1311, 12nfxp 5553 . . . 4 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
14 sneq 4535 . . . . 5 (𝑥 = 𝑢 → {𝑥} = {𝑢})
15 csbeq1a 3842 . . . . 5 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
1614, 15xpeq12d 5551 . . . 4 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
1710, 13, 16cbviun 4924 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
1817mpteq1i 5121 . 2 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶) = (𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
19 nfcv 2955 . . 3 𝑢𝐵
20 nfcv 2955 . . 3 𝑢𝐶
21 nfcv 2955 . . 3 𝑣𝐶
22 nfcsb1v 3852 . . 3 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
23 nfcv 2955 . . . 4 𝑦𝑢
24 nfcsb1v 3852 . . . 4 𝑦𝑣 / 𝑦𝐶
2523, 24nfcsbw 3854 . . 3 𝑦𝑢 / 𝑥𝑣 / 𝑦𝐶
26 csbeq1a 3842 . . . 4 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
27 csbeq1a 3842 . . . 4 (𝑥 = 𝑢𝑣 / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2826, 27sylan9eqr 2855 . . 3 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2919, 12, 20, 21, 22, 25, 15, 28cbvmpox 7227 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
309, 18, 293eqtr4ri 2832 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ⦋csb 3828  {csn 4525  ⟨cop 4531  ∪ ciun 4882   ↦ cmpt 5111   × cxp 5518  ‘cfv 6325   ∈ cmpo 7138  1st c1st 7672  2nd c2nd 7673 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-iota 6284  df-fun 6327  df-fv 6333  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675 This theorem is referenced by:  mpompts  7748  ovmptss  7774
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