Theorem dfmpo 8143

Description: Alternate definition for the maps-to notation df-mpo 7453 (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 8106	. 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 5329	. . . 4 ⊢ ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V
4	3	csbex 5329	. . 3 ⊢ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V
5	4	dfmpt 7178	. 2 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶) = ∪ 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
6		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑥𝑤
7		nfcsb1v 3946	. . . . 5 ⊢ Ⅎ𝑥⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶
8	6, 7	nfop 4913	. . . 4 ⊢ Ⅎ𝑥⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩
9	8	nfsn 4732	. . 3 ⊢ Ⅎ𝑥{⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
10		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑦𝑤
11		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑦(1st ‘𝑤)
12		nfcsb1v 3946	. . . . . 6 ⊢ Ⅎ𝑦⦋(2nd ‘𝑤) / 𝑦⦌𝐶
13	11, 12	nfcsbw 3948	. . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶
14	10, 13	nfop 4913	. . . 4 ⊢ Ⅎ𝑦⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩
15	14	nfsn 4732	. . 3 ⊢ Ⅎ𝑦{⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩}
16		nfcv 2908	. . 3 ⊢ Ⅎ𝑤{⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
17		id 22	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → 𝑤 = ⟨𝑥, 𝑦⟩)
18		csbopeq1a 8091	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 = 𝐶)
19	17, 18	opeq12d 4905	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩ = ⟨⟨𝑥, 𝑦⟩, 𝐶⟩)
20	19	sneqd 4660	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → {⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩} = {⟨⟨𝑥, 𝑦⟩, 𝐶⟩})
21	9, 15, 16, 20	iunxpf 5873	. 2 ⊢ ∪ 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶⟩} = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
22	1, 5, 21	3eqtri 2772	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}

Syntax hints:   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ⦋csb 3921  {csn 4648  ⟨cop 4654  ∪ ciun 5015   ↦ cmpt 5249   × cxp 5698  ‘cfv 6573   ∈ cmpo 7450  1^st c1st 8028  2^nd c2nd 8029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031

This theorem is referenced by:  fpar  8157