Theorem dfmpo 8128

Description: Alternate definition for the maps-to notation df-mpo 7437 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
dfmpo.1	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
dfmpo	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem dfmpo

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpompts 8091	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶)
2		dfmpo.1	. . . . 5 ⊢ 𝐶 ∈ V
3	2	csbex 5310	. . . 4 ⊢ ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V
4	3	csbex 5310	. . 3 ⊢ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V
5	4	dfmpt 7163	. 2 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶) = ∪ 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
6		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑥𝑤
7		nfcsb1v 3922	. . . . 5 ⊢ Ⅎ𝑥⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶
8	6, 7	nfop 4888	. . . 4 ⊢ Ⅎ𝑥⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩
9	8	nfsn 4706	. . 3 ⊢ Ⅎ𝑥{⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
10		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑦𝑤
11		nfcv 2904	. . . . . 6 ⊢ Ⅎ𝑦(1st ‘𝑤)
12		nfcsb1v 3922	. . . . . 6 ⊢ Ⅎ𝑦⦋(2nd ‘𝑤) / 𝑦⦌𝐶
13	11, 12	nfcsbw 3924	. . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶
14	10, 13	nfop 4888	. . . 4 ⊢ Ⅎ𝑦⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩
15	14	nfsn 4706	. . 3 ⊢ Ⅎ𝑦{⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
16		nfcv 2904	. . 3 ⊢ Ⅎ𝑤{⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
17		id 22	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → 𝑤 = ⟨𝑥, 𝑦⟩)
18		csbopeq1a 8076	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 = 𝐶)
19	17, 18	opeq12d 4880	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩ = ⟨⟨𝑥, 𝑦⟩, 𝐶⟩)
20	19	sneqd 4637	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → {⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩} = {⟨⟨𝑥, 𝑦⟩, 𝐶⟩})
21	9, 15, 16, 20	iunxpf 5858	. 2 ⊢ ∪ 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩} = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
22	1, 5, 21	3eqtri 2768	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}

Syntax hints:   = wceq 1539   ∈ wcel 2107  Vcvv 3479  ⦋csb 3898  {csn 4625  ⟨cop 4631  ∪ ciun 4990   ↦ cmpt 5224   × cxp 5682  ‘cfv 6560   ∈ cmpo 7434  1^st c1st 8013  2^nd c2nd 8014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016

This theorem is referenced by:  fpar  8142