Theorem mpomptsx 8062

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 3459	. . . . . 6 ⊢ 𝑢 ∈ V
2		vex 3459	. . . . . 6 ⊢ 𝑣 ∈ V
3	1, 2	op1std 7997	. . . . 5 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (1st ‘𝑧) = 𝑢)
4	3	csbeq1d 3858	. . . 4 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)
5	1, 2	op2ndd 7998	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (2nd ‘𝑧) = 𝑣)
6	5	csbeq1d 3858	. . . . 5 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
7	6	csbeq2dv 3861	. . . 4 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋𝑢 / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
8	4, 7	eqtrd 2798	. . 3 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
9	8	mpomptx 7525	. 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 2925	. . . 4 ⊢ Ⅎ𝑢({𝑥} × 𝐵)
11		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑥{𝑢}
12		nfcsb1v 3878	. . . . 5 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵
13	11, 12	nfxp 5696	. . . 4 ⊢ Ⅎ𝑥({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
14		sneq 4600	. . . . 5 ⊢ (𝑥 = 𝑢 → {𝑥} = {𝑢})
15		csbeq1a 3868	. . . . 5 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵)
16	14, 15	xpeq12d 5694	. . . 4 ⊢ (𝑥 = 𝑢 → ({𝑥} × 𝐵) = ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵))
17	10, 13, 16	cbviun 5000	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = ∪ 𝑢 ∈ 𝐴 ({𝑢} × ⦋𝑢 / 𝑥⦌𝐵)
18	17	mpteq1i 5203	. 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 2925	. . 3 ⊢ Ⅎ𝑢𝐵
20		nfcv 2925	. . 3 ⊢ Ⅎ𝑢𝐶
21		nfcv 2925	. . 3 ⊢ Ⅎ𝑣𝐶
22		nfcsb1v 3878	. . 3 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
23		nfcv 2925	. . . 4 ⊢ Ⅎ𝑦𝑢
24		nfcsb1v 3878	. . . 4 ⊢ Ⅎ𝑦⦋𝑣 / 𝑦⦌𝐶
25	23, 24	nfcsbw 3880	. . 3 ⊢ Ⅎ𝑦⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶
26		csbeq1a 3868	. . . 4 ⊢ (𝑦 = 𝑣 → 𝐶 = ⦋𝑣 / 𝑦⦌𝐶)
27		csbeq1a 3868	. . . 4 ⊢ (𝑥 = 𝑢 → ⦋𝑣 / 𝑦⦌𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
28	26, 27	sylan9eqr 2820	. . 3 ⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝐶 = ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
29	19, 12, 20, 21, 22, 25, 15, 28	cbvmpox 7505	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑢 ∈ 𝐴, 𝑣 ∈ ⦋𝑢 / 𝑥⦌𝐵 ↦ ⦋𝑢 / 𝑥⦌⦋𝑣 / 𝑦⦌𝐶)
30	9, 18, 29	3eqtr4ri 2797	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶)

Syntax hints:   = wceq 1570  ⦋csb 3854  {csn 4590  ⟨cop 4596  ∪ ciun 4957   ↦ cmpt 5193   × cxp 5661  ‘cfv 6538   ∈ cmpo 7414  1^st c1st 7985  2^nd c2nd 7986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fv 6546  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988

This theorem is referenced by:  mpompts  8063  ovmptss  8089