Theorem dfdm5 35790

Description: Definition of domain in terms of 1^st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

Assertion

Ref	Expression
dfdm5	⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)

Proof of Theorem dfdm5

Dummy variables 𝑝 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		excom 2162	. . . 4 ⊢ (∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
2		opex 5439	. . . . . . . 8 ⊢ ⟨𝑧, 𝑦⟩ ∈ V
3		breq1 5122	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝1st 𝑥 ↔ ⟨𝑧, 𝑦⟩1st 𝑥))
4		eleq1 2822	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝 ∈ 𝐴 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
5	3, 4	anbi12d 632	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴) ↔ (⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
6		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
7		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
8	6, 7	br1steq 35788	. . . . . . . . . . 11 ⊢ (⟨𝑧, 𝑦⟩1st 𝑥 ↔ 𝑥 = 𝑧)
9		equcom 2017	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
10	8, 9	bitri 275	. . . . . . . . . 10 ⊢ (⟨𝑧, 𝑦⟩1st 𝑥 ↔ 𝑧 = 𝑥)
11	10	anbi1i 624	. . . . . . . . 9 ⊢ ((⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
12	5, 11	bitrdi 287	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
13	2, 12	ceqsexv 3511	. . . . . . 7 ⊢ (∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
14	13	exbii 1848	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
15		excom 2162	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
16		vex 3463	. . . . . . 7 ⊢ 𝑥 ∈ V
17		opeq1 4849	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
18	17	eleq1d 2819	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
19	16, 18	ceqsexv 3511	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
20	14, 15, 19	3bitr3ri 302	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
21	20	exbii 1848	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
22		ancom 460	. . . . . 6 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥) ↔ (𝑝(1st ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴))
23		anass 468	. . . . . . 7 ⊢ (((∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
24	16	brresi 5975	. . . . . . . . 9 ⊢ (𝑝(1st ↾ (V × V))𝑥 ↔ (𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥))
25		elvv 5729	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (V × V) ↔ ∃𝑧∃𝑦 𝑝 = ⟨𝑧, 𝑦⟩)
26		excom 2162	. . . . . . . . . . 11 ⊢ (∃𝑧∃𝑦 𝑝 = ⟨𝑧, 𝑦⟩ ↔ ∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
27	25, 26	bitri 275	. . . . . . . . . 10 ⊢ (𝑝 ∈ (V × V) ↔ ∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
28	27	anbi1i 624	. . . . . . . . 9 ⊢ ((𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
29	24, 28	bitri 275	. . . . . . . 8 ⊢ (𝑝(1st ↾ (V × V))𝑥 ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
30	29	anbi1i 624	. . . . . . 7 ⊢ ((𝑝(1st ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴) ↔ ((∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝 ∈ 𝐴))
31		19.41vv 1950	. . . . . . 7 ⊢ (∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
32	23, 30, 31	3bitr4i 303	. . . . . 6 ⊢ ((𝑝(1st ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴) ↔ ∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
33	22, 32	bitri 275	. . . . 5 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
34	33	exbii 1848	. . . 4 ⊢ (∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑝∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
35	1, 21, 34	3bitr4i 303	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥))
36	16	eldm2 5881	. . 3 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
37	16	elima2 6053	. . 3 ⊢ (𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴) ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥))
38	35, 36, 37	3bitr4i 303	. 2 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴))
39	38	eqriv 2732	1 ⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3459  ⟨cop 4607   class class class wbr 5119   × cxp 5652  dom cdm 5654   ↾ cres 5656   “ cima 5657  1^st c1st 7986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537  df-fv 6539  df-1st 7988

This theorem is referenced by:  brdomain  35951