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Theorem mpt2mptsx 7178
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 (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 mpt2mptsx
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3189 . . . . . 6 𝑢 ∈ V
2 vex 3189 . . . . . 6 𝑣 ∈ V
31, 2op1std 7123 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) = 𝑢)
43csbeq1d 3521 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶)
51, 2op2ndd 7124 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) = 𝑣)
65csbeq1d 3521 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
76csbeq2dv 3964 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
84, 7eqtrd 2655 . . 3 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
98mpt2mptx 6704 . 2 (𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
10 nfcv 2761 . . . 4 𝑢({𝑥} × 𝐵)
11 nfcv 2761 . . . . 5 𝑥{𝑢}
12 nfcsb1v 3530 . . . . 5 𝑥𝑢 / 𝑥𝐵
1311, 12nfxp 5102 . . . 4 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
14 sneq 4158 . . . . 5 (𝑥 = 𝑢 → {𝑥} = {𝑢})
15 csbeq1a 3523 . . . . 5 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
1614, 15xpeq12d 5100 . . . 4 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
1710, 13, 16cbviun 4523 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
18 mpteq1 4697 . . 3 ( 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) → (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶) = (𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶))
1917, 18ax-mp 5 . 2 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶) = (𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
20 nfcv 2761 . . 3 𝑢𝐵
21 nfcv 2761 . . 3 𝑢𝐶
22 nfcv 2761 . . 3 𝑣𝐶
23 nfcsb1v 3530 . . 3 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
24 nfcv 2761 . . . 4 𝑦𝑢
25 nfcsb1v 3530 . . . 4 𝑦𝑣 / 𝑦𝐶
2624, 25nfcsb 3532 . . 3 𝑦𝑢 / 𝑥𝑣 / 𝑦𝐶
27 csbeq1a 3523 . . . 4 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
28 csbeq1a 3523 . . . 4 (𝑥 = 𝑢𝑣 / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2927, 28sylan9eqr 2677 . . 3 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
3020, 12, 21, 22, 23, 26, 15, 29cbvmpt2x 6686 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
319, 19, 303eqtr4ri 2654 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  csb 3514  {csn 4148  cop 4154   ciun 4485  cmpt 4673   × cxp 5072  cfv 5847  cmpt2 6606  1st c1st 7111  2nd c2nd 7112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114
This theorem is referenced by:  mpt2mpts  7179  ovmptss  7203
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