Theorem dfmpo 8113

Description: Alternate definition for the maps-to notation df-mpo 7425 (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 8076	. 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 5264	. . . 4 ⊢ ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V
4	3	csbex 5264	. . 3 ⊢ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V
5	4	dfmpt 7147	. 2 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶) = ∪ 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
6		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑥𝑤
7		nfcsb1v 3870	. . . . 5 ⊢ Ⅎ𝑥⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶
8	6, 7	nfop 4848	. . . 4 ⊢ Ⅎ𝑥⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩
9	8	nfsn 4667	. . 3 ⊢ Ⅎ𝑥{⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
10		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑦𝑤
11		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑦(1st ‘𝑤)
12		nfcsb1v 3870	. . . . . 6 ⊢ Ⅎ𝑦⦋(2nd ‘𝑤) / 𝑦⦌𝐶
13	11, 12	nfcsbw 3872	. . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶
14	10, 13	nfop 4848	. . . 4 ⊢ Ⅎ𝑦⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩
15	14	nfsn 4667	. . 3 ⊢ Ⅎ𝑦{⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
16		nfcv 2922	. . 3 ⊢ Ⅎ𝑤{⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
17		id 23	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → 𝑤 = ⟨𝑥, 𝑦⟩)
18		csbopeq1a 8061	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 = 𝐶)
19	17, 18	opeq12d 4840	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩ = ⟨⟨𝑥, 𝑦⟩, 𝐶⟩)
20	19	sneqd 4595	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → {⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩} = {⟨⟨𝑥, 𝑦⟩, 𝐶⟩})
21	9, 15, 16, 20	iunxpf 5825	. 2 ⊢ ∪ 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩} = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
22	1, 5, 21	3eqtri 2787	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⦋csb 3846  {csn 4583  ⟨cop 4589  ∪ ciun 4950   ↦ cmpt 5185   × cxp 5648  ‘cfv 6538   ∈ cmpo 7422  1^st c1st 7999  2^nd c2nd 8000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002

This theorem is used by:  fpar  8127