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Theorem mpompts 6428

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 6427	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1^st ‘𝑧) / 𝑥⦌⦋(2^nd ‘𝑧) / 𝑦⦌𝐶)
2		iunxpconst 4833	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = (𝐴 × 𝐵)
3		mpteq1 4213	. . 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 2259	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 1402 ⦋csb 3147 {csn 3708 ∪ ciun 4010 ↦ cmpt 4190 × cxp 4770 ‘cfv 5375 ∈ cmpo 6081 1^st c1st 6366 2^nd c2nd 6367

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 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4247 ax-pow 4309 ax-pr 4344 ax-un 4576

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-v 2823 df-sbc 3052 df-csb 3148 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-id 4436 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-iota 5335 df-fun 5377 df-fv 5383 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369

This theorem is referenced by: dfmpo 6453