Theorem dfmpo 6249

Description: Alternate definition for the maps-to notation df-mpo 5902 (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 6224	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶)
2		vex 2755	. . . . 5 ⊢ 𝑤 ∈ V
3		1stexg 6193	. . . . 5 ⊢ (𝑤 ∈ V → (1st ‘𝑤) ∈ V)
4	2, 3	ax-mp 5	. . . 4 ⊢ (1st ‘𝑤) ∈ V
5		2ndexg 6194	. . . . . 6 ⊢ (𝑤 ∈ V → (2nd ‘𝑤) ∈ V)
6	2, 5	ax-mp 5	. . . . 5 ⊢ (2nd ‘𝑤) ∈ V
7		dfmpo.1	. . . . 5 ⊢ 𝐶 ∈ V
8	6, 7	csbexa 4147	. . . 4 ⊢ ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V
9	4, 8	csbexa 4147	. . 3 ⊢ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V
10	9	dfmpt 5714	. 2 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶) = ∪ 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
11		nfcv 2332	. . . . 5 ⊢ Ⅎ𝑥𝑤
12		nfcsb1v 3105	. . . . 5 ⊢ Ⅎ𝑥⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶
13	11, 12	nfop 3809	. . . 4 ⊢ Ⅎ𝑥⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩
14	13	nfsn 3667	. . 3 ⊢ Ⅎ𝑥{⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
15		nfcv 2332	. . . . 5 ⊢ Ⅎ𝑦𝑤
16		nfcv 2332	. . . . . 6 ⊢ Ⅎ𝑦(1st ‘𝑤)
17		nfcsb1v 3105	. . . . . 6 ⊢ Ⅎ𝑦⦋(2nd ‘𝑤) / 𝑦⦌𝐶
18	16, 17	nfcsb 3109	. . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶
19	15, 18	nfop 3809	. . . 4 ⊢ Ⅎ𝑦⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩
20	19	nfsn 3667	. . 3 ⊢ Ⅎ𝑦{⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
21		nfcv 2332	. . 3 ⊢ Ⅎ𝑤{⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
22		id 19	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → 𝑤 = ⟨𝑥, 𝑦⟩)
23		csbopeq1a 6214	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 = 𝐶)
24	22, 23	opeq12d 3801	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩ = ⟨⟨𝑥, 𝑦⟩, 𝐶⟩)
25	24	sneqd 3620	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → {⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩} = {⟨⟨𝑥, 𝑦⟩, 𝐶⟩})
26	14, 20, 21, 25	iunxpf 4793	. 2 ⊢ ∪ 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩} = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
27	1, 10, 26	3eqtri 2214	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}

Syntax hints:   = wceq 1364   ∈ wcel 2160  Vcvv 2752  ⦋csb 3072  {csn 3607  ⟨cop 3610  ∪ ciun 3901   ↦ cmpt 4079   × cxp 4642  ‘cfv 5235   ∈ cmpo 5899  1^st c1st 6164  2^nd c2nd 6165

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167

This theorem is referenced by: (None)