Theorem dfdm5 34030

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 5413	. . . . . . . 8 ⊢ ⟨𝑧, 𝑦⟩ ∈ V
3		breq1 5099	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝1st 𝑥 ↔ ⟨𝑧, 𝑦⟩1st 𝑥))
4		eleq1 2825	. . . . . . . . . 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 3446	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
7		vex 3446	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
8	6, 7	br1steq 34028	. . . . . . . . . . 11 ⊢ (⟨𝑧, 𝑦⟩1st 𝑥 ↔ 𝑥 = 𝑧)
9		equcom 2021	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
10	8, 9	bitri 275	. . . . . . . . . 10 ⊢ (⟨𝑧, 𝑦⟩1st 𝑥 ↔ 𝑧 = 𝑥)
11	10	anbi1i 625	. . . . . . . . 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 3490	. . . . . . 7 ⊢ (∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
14	13	exbii 1850	. . . . . 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 3446	. . . . . . 7 ⊢ 𝑥 ∈ V
17		opeq1 4821	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
18	17	eleq1d 2822	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
19	16, 18	ceqsexv 3490	. . . . . 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 1850	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
22		ancom 462	. . . . . 6 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥) ↔ (𝑝(1st ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴))
23		anass 470	. . . . . . 7 ⊢ (((∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
24	16	brresi 5936	. . . . . . . . 9 ⊢ (𝑝(1st ↾ (V × V))𝑥 ↔ (𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥))
25		elvv 5696	. . . . . . . . . . 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 625	. . . . . . . . 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 625	. . . . . . 7 ⊢ ((𝑝(1st ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴) ↔ ((∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝 ∈ 𝐴))
31		19.41vv 1954	. . . . . . 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 1850	. . . 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 5847	. . 3 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
37	16	elima2 6009	. . 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 2734	1 ⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)

Syntax hints:   ∧ wa 397   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3442  ⟨cop 4583   class class class wbr 5096   × cxp 5622  dom cdm 5624   ↾ cres 5626   “ cima 5627  1^st c1st 7901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5247  ax-nul 5254  ax-pr 5376  ax-un 7654

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-br 5097  df-opab 5159  df-mpt 5180  df-id 5522  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-fo 6489  df-fv 6491  df-1st 7903

This theorem is referenced by:  brdomain  34372