Theorem mpomptsx 8010

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 3434	. . . . . 6 ⊢ 𝑢 ∈ V
2		vex 3434	. . . . . 6 ⊢ 𝑣 ∈ V
3	1, 2	op1std 7945	. . . . 5 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑧) = 𝑢)
4	3	csbeq1d 3842	. . . 4 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
5	1, 2	op2ndd 7946	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑧) = 𝑣)
6	5	csbeq1d 3842	. . . . 5 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
7	6	csbeq2dv 3845	. . . 4 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
8	4, 7	eqtrd 2772	. . 3 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
9	8	mpomptx 7473	. 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 2899	. . . 4 ⊢ Ⅎ𝑢({𝑥} × 𝐵)
11		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑥{𝑢}
12		nfcsb1v 3862	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵
13	11, 12	nfxp 5657	. . . 4 ⊢ Ⅎ𝑥({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
14		sneq 4578	. . . . 5 ⊢ (𝑥 = 𝑢 → {𝑥} = {𝑢})
15		csbeq1a 3852	. . . . 5 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵)
16	14, 15	xpeq12d 5655	. . . 4 ⊢ (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵))
17	10, 13, 16	cbviun 4978	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
18	17	mpteq1i 5177	. 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 2899	. . 3 ⊢ Ⅎ𝑢𝐵
20		nfcv 2899	. . 3 ⊢ Ⅎ𝑢𝐶
21		nfcv 2899	. . 3 ⊢ Ⅎ𝑣𝐶
22		nfcsb1v 3862	. . 3 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
23		nfcv 2899	. . . 4 ⊢ Ⅎ𝑦𝑢
24		nfcsb1v 3862	. . . 4 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝐶
25	23, 24	nfcsbw 3864	. . 3 ⊢ Ⅎ𝑦⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
26		csbeq1a 3852	. . . 4 ⊢ (𝑦 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
27		csbeq1a 3852	. . . 4 ⊢ (𝑥 = 𝑢 → ⦋𝑣 / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
28	26, 27	sylan9eqr 2794	. . 3 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
29	19, 12, 20, 21, 22, 25, 15, 28	cbvmpox 7453	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑢 ∈ 𝐴, 𝑣 ∈ ⦋𝑢 / 𝑥⦌𝐵 ↦ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
30	9, 18, 29	3eqtr4ri 2771	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)

Syntax hints:   = wceq 1542  ⦋csb 3838  {csn 4568  ⟨cop 4574  ∪ ciun 4934   ↦ cmpt 5167   × cxp 5622  ‘cfv 6492   ∈ cmpo 7362  1^st c1st 7933  2^nd c2nd 7934

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936

This theorem is referenced by:  mpompts  8011  ovmptss  8036