Theorem mpomptsx 8063

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	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)

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.

Step	Hyp	Ref	Expression
1		vex 3463	. . . . . 6 ⊢ 𝑢 ∈ V
2		vex 3463	. . . . . 6 ⊢ 𝑣 ∈ V
3	1, 2	op1std 7998	. . . . 5 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑧) = 𝑢)
4	3	csbeq1d 3878	. . . 4 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
5	1, 2	op2ndd 7999	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑧) = 𝑣)
6	5	csbeq1d 3878	. . . . 5 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
7	6	csbeq2dv 3881	. . . 4 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
8	4, 7	eqtrd 2770	. . 3 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
9	8	mpomptx 7520	. 2 ⊢ (𝑧 ∈ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶) = (𝑢 ∈ 𝐴, 𝑣 ∈ ⦋𝑢 / 𝑥⦌𝐵 ↦ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
10		nfcv 2898	. . . 4 ⊢ Ⅎ𝑢({𝑥} × 𝐵)
11		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥{𝑢}
12		nfcsb1v 3898	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵
13	11, 12	nfxp 5687	. . . 4 ⊢ Ⅎ𝑥({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
14		sneq 4611	. . . . 5 ⊢ (𝑥 = 𝑢 → {𝑥} = {𝑢})
15		csbeq1a 3888	. . . . 5 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵)
16	14, 15	xpeq12d 5685	. . . 4 ⊢ (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵))
17	10, 13, 16	cbviun 5012	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
18	17	mpteq1i 5211	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶) = (𝑧 ∈ ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
19		nfcv 2898	. . 3 ⊢ Ⅎ𝑢𝐵
20		nfcv 2898	. . 3 ⊢ Ⅎ𝑢𝐶
21		nfcv 2898	. . 3 ⊢ Ⅎ𝑣𝐶
22		nfcsb1v 3898	. . 3 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
23		nfcv 2898	. . . 4 ⊢ Ⅎ𝑦𝑢
24		nfcsb1v 3898	. . . 4 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝐶
25	23, 24	nfcsbw 3900	. . 3 ⊢ Ⅎ𝑦⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
26		csbeq1a 3888	. . . 4 ⊢ (𝑦 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
27		csbeq1a 3888	. . . 4 ⊢ (𝑥 = 𝑢 → ⦋𝑣 / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
28	26, 27	sylan9eqr 2792	. . 3 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
29	19, 12, 20, 21, 22, 25, 15, 28	cbvmpox 7500	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑢 ∈ 𝐴, 𝑣 ∈ ⦋𝑢 / 𝑥⦌𝐵 ↦ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
30	9, 18, 29	3eqtr4ri 2769	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)

Syntax hints:   = wceq 1540  ⦋csb 3874  {csn 4601  ⟨cop 4607  ∪ ciun 4967   ↦ cmpt 5201   × cxp 5652  ‘cfv 6531   ∈ cmpo 7407  1^st c1st 7986  2^nd c2nd 7987

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fv 6539  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989

This theorem is referenced by:  mpompts  8064  ovmptss  8092