Theorem dfmpo 6120

Description: Alternate definition for the maps-to notation df-mpo 5779 (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 6096	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶)
2		vex 2689	. . . . 5 ⊢ 𝑤 ∈ V
3		1stexg 6065	. . . . 5 ⊢ (𝑤 ∈ V → (1st ‘𝑤) ∈ V)
4	2, 3	ax-mp 5	. . . 4 ⊢ (1st ‘𝑤) ∈ V
5		2ndexg 6066	. . . . . 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 4057	. . . 4 ⊢ ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V
9	4, 8	csbexa 4057	. . 3 ⊢ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V
10	9	dfmpt 5597	. 2 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶) = ∪ 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
11		nfcv 2281	. . . . 5 ⊢ Ⅎ𝑥𝑤
12		nfcsb1v 3035	. . . . 5 ⊢ Ⅎ𝑥⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶
13	11, 12	nfop 3721	. . . 4 ⊢ Ⅎ𝑥⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩
14	13	nfsn 3583	. . 3 ⊢ Ⅎ𝑥{⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
15		nfcv 2281	. . . . 5 ⊢ Ⅎ𝑦𝑤
16		nfcv 2281	. . . . . 6 ⊢ Ⅎ𝑦(1st ‘𝑤)
17		nfcsb1v 3035	. . . . . 6 ⊢ Ⅎ𝑦⦋(2nd ‘𝑤) / 𝑦⦌𝐶
18	16, 17	nfcsb 3037	. . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶
19	15, 18	nfop 3721	. . . 4 ⊢ Ⅎ𝑦⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩
20	19	nfsn 3583	. . 3 ⊢ Ⅎ𝑦{⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
21		nfcv 2281	. . 3 ⊢ Ⅎ𝑤{⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
22		id 19	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → 𝑤 = ⟨𝑥, 𝑦⟩)
23		csbopeq1a 6086	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 = 𝐶)
24	22, 23	opeq12d 3713	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩ = ⟨⟨𝑥, 𝑦⟩, 𝐶⟩)
25	24	sneqd 3540	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → {⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩} = {⟨⟨𝑥, 𝑦⟩, 𝐶⟩})
26	14, 20, 21, 25	iunxpf 4687	. 2 ⊢ ∪ 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩} = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
27	1, 10, 26	3eqtri 2164	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}

Syntax hints:   = wceq 1331   ∈ wcel 1480  Vcvv 2686  ⦋csb 3003  {csn 3527  ⟨cop 3530  ∪ ciun 3813   ↦ cmpt 3989   × cxp 4537  ‘cfv 5123   ∈ cmpo 5776  1^st c1st 6036  2^nd c2nd 6037

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039

This theorem is referenced by: (None)