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Theorem mpompts 6146

Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.)

**Assertion**
Ref	Expression
mpompts	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶)

Distinct variable groups: 𝑥,𝑦,𝑧,𝐴 𝑦,𝐵,𝑧 𝑧,𝐶 𝑥,𝐵 Allowed substitution hints: 𝐶(𝑥,𝑦)

**Proof of Theorem mpompts**
Step	Hyp	Ref	Expression
1		mpomptsx 6145	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶)
2		iunxpconst 4646	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
3		mpteq1 4048	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵) → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶))
4	2, 3	ax-mp 5	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶)
5	1, 4	eqtri 2178	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶)

Colors of variables: wff set class

Syntax hints: = wceq 1335 ⦋csb 3031 {csn 3560 ∪ ciun 3849 ↦ cmpt 4025 × cxp 4584 ‘cfv 5170 ∈ cmpo 5826 1^st c1st 6086 2^nd c2nd 6087

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-iota 5135 df-fun 5172 df-fv 5178 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089

This theorem is referenced by: dfmpo 6170