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Theorem mpt2mptsx 7393
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 3335 . . . . . 6 𝑢 ∈ V
2 vex 3335 . . . . . 6 𝑣 ∈ V
31, 2op1std 7335 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) = 𝑢)
43csbeq1d 3673 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶)
51, 2op2ndd 7336 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) = 𝑣)
65csbeq1d 3673 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd𝑧) / 𝑦𝐶 = 𝑣 / 𝑦𝐶)
76csbeq2dv 4127 . . . 4 (𝑧 = ⟨𝑢, 𝑣⟩ → 𝑢 / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
84, 7eqtrd 2786 . . 3 (𝑧 = ⟨𝑢, 𝑣⟩ → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
98mpt2mptx 6908 . 2 (𝑧 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
10 nfcv 2894 . . . 4 𝑢({𝑥} × 𝐵)
11 nfcv 2894 . . . . 5 𝑥{𝑢}
12 nfcsb1v 3682 . . . . 5 𝑥𝑢 / 𝑥𝐵
1311, 12nfxp 5291 . . . 4 𝑥({𝑢} × 𝑢 / 𝑥𝐵)
14 sneq 4323 . . . . 5 (𝑥 = 𝑢 → {𝑥} = {𝑢})
15 csbeq1a 3675 . . . . 5 (𝑥 = 𝑢𝐵 = 𝑢 / 𝑥𝐵)
1614, 15xpeq12d 5289 . . . 4 (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × 𝑢 / 𝑥𝐵))
1710, 13, 16cbviun 4701 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = 𝑢𝐴 ({𝑢} × 𝑢 / 𝑥𝐵)
18 mpteq1 4881 . . 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 2894 . . 3 𝑢𝐵
21 nfcv 2894 . . 3 𝑢𝐶
22 nfcv 2894 . . 3 𝑣𝐶
23 nfcsb1v 3682 . . 3 𝑥𝑢 / 𝑥𝑣 / 𝑦𝐶
24 nfcv 2894 . . . 4 𝑦𝑢
25 nfcsb1v 3682 . . . 4 𝑦𝑣 / 𝑦𝐶
2624, 25nfcsb 3684 . . 3 𝑦𝑢 / 𝑥𝑣 / 𝑦𝐶
27 csbeq1a 3675 . . . 4 (𝑦 = 𝑣𝐶 = 𝑣 / 𝑦𝐶)
28 csbeq1a 3675 . . . 4 (𝑥 = 𝑢𝑣 / 𝑦𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
2927, 28sylan9eqr 2808 . . 3 ((𝑥 = 𝑢𝑦 = 𝑣) → 𝐶 = 𝑢 / 𝑥𝑣 / 𝑦𝐶)
3020, 12, 21, 22, 23, 26, 15, 29cbvmpt2x 6890 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑢𝐴, 𝑣𝑢 / 𝑥𝐵𝑢 / 𝑥𝑣 / 𝑦𝐶)
319, 19, 303eqtr4ri 2785 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1624  csb 3666  {csn 4313  cop 4319   ciun 4664  cmpt 4873   × cxp 5256  cfv 6041  cmpt2 6807  1st c1st 7323  2nd c2nd 7324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fv 6049  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326
This theorem is referenced by:  mpt2mpts  7394  ovmptss  7418
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