Theorem dfdm5 36136

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 2199	. . . 4 ⊢ (∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
2		opex 5436	. . . . . . . 8 ⊢ ⟨𝑧, 𝑦⟩ ∈ V
3		breq1 5108	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝1st 𝑥 ↔ ⟨𝑧, 𝑦⟩1st 𝑥))
4		eleq1 2853	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝 ∈ 𝐴 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
5	3, 4	anbi12d 643	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴) ↔ (⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
6		vex 3461	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
7		vex 3461	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
8	6, 7	br1steq 36134	. . . . . . . . . . 11 ⊢ (⟨𝑧, 𝑦⟩1st 𝑥 ↔ 𝑥 = 𝑧)
9		equcom 2041	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
10	8, 9	bitri 278	. . . . . . . . . 10 ⊢ (⟨𝑧, 𝑦⟩1st 𝑥 ↔ 𝑧 = 𝑥)
11	10	anbi1i 635	. . . . . . . . 9 ⊢ ((⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
12	5, 11	bitrdi 290	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
13	2, 12	ceqsexv 3505	. . . . . . 7 ⊢ (∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
14	13	exbii 1871	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
15		excom 2199	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
16		vex 3461	. . . . . . 7 ⊢ 𝑥 ∈ V
17		opeq1 4834	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
18	17	eleq1d 2850	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
19	16, 18	ceqsexv 3505	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
20	14, 15, 19	3bitr3ri 305	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
21	20	exbii 1871	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
22		ancom 465	. . . . . 6 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥) ↔ (𝑝(1st ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴))
23		anass 473	. . . . . . 7 ⊢ (((∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
24	16	brresi 5978	. . . . . . . . 9 ⊢ (𝑝(1st ↾ (V × V))𝑥 ↔ (𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥))
25		elvv 5727	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (V × V) ↔ ∃𝑧∃𝑦 𝑝 = ⟨𝑧, 𝑦⟩)
26		excom 2199	. . . . . . . . . . 11 ⊢ (∃𝑧∃𝑦 𝑝 = ⟨𝑧, 𝑦⟩ ↔ ∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
27	25, 26	bitri 278	. . . . . . . . . 10 ⊢ (𝑝 ∈ (V × V) ↔ ∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
28	27	anbi1i 635	. . . . . . . . 9 ⊢ ((𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
29	24, 28	bitri 278	. . . . . . . 8 ⊢ (𝑝(1st ↾ (V × V))𝑥 ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
30	29	anbi1i 635	. . . . . . 7 ⊢ ((𝑝(1st ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴) ↔ ((∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝 ∈ 𝐴))
31		19.41vv 1973	. . . . . . 7 ⊢ (∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
32	23, 30, 31	3bitr4i 306	. . . . . 6 ⊢ ((𝑝(1st ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴) ↔ ∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
33	22, 32	bitri 278	. . . . 5 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
34	33	exbii 1871	. . . 4 ⊢ (∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑝∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
35	1, 21, 34	3bitr4i 306	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥))
36	16	eldm2 5882	. . 3 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
37	16	elima2 6059	. . 3 ⊢ (𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴) ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥))
38	35, 36, 37	3bitr4i 306	. 2 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴))
39	38	eqriv 2762	1 ⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)

Syntax hints:   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  Vcvv 3457  ⟨cop 4591   class class class wbr 5105   × cxp 5650  dom cdm 5652   ↾ cres 5654   “ cima 5655  1^st c1st 7972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fo 6531  df-fv 6533  df-1st 7974

This theorem is referenced by:  brdomain  36294