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Theorem mpompts 6294

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 6293	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶)
2		iunxpconst 4740	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
3		mpteq1 4133	. . 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 2227	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 1373 ⦋csb 3095 {csn 3635 ∪ ciun 3930 ↦ cmpt 4110 × cxp 4678 ‘cfv 5277 ∈ cmpo 5956 1^st c1st 6234 2^nd c2nd 6235

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-sep 4167 ax-pow 4223 ax-pr 4258 ax-un 4485

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ral 2490 df-rex 2491 df-v 2775 df-sbc 3001 df-csb 3096 df-un 3172 df-in 3174 df-ss 3181 df-pw 3620 df-sn 3641 df-pr 3642 df-op 3644 df-uni 3854 df-iun 3932 df-br 4049 df-opab 4111 df-mpt 4112 df-id 4345 df-xp 4686 df-rel 4687 df-cnv 4688 df-co 4689 df-dm 4690 df-rn 4691 df-iota 5238 df-fun 5279 df-fv 5285 df-oprab 5958 df-mpo 5959 df-1st 6236 df-2nd 6237

This theorem is referenced by: dfmpo 6319