Theorem dfdm5 30921

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.)

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 2029	. . . 4 ⊢ (∃𝑦∃𝑝∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
2		opex 4859	. . . . . . . 8 ⊢ ⟨𝑧, 𝑦⟩ ∈ V
3		breq1 4586	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝1st 𝑥 ↔ ⟨𝑧, 𝑦⟩1st 𝑥))
4		eleq1 2676	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝 ∈ 𝐴 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
5	3, 4	anbi12d 743	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴) ↔ (⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
6		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
7		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
8		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
9	6, 7, 8	br1steq 30917	. . . . . . . . . . 11 ⊢ (⟨𝑧, 𝑦⟩1st 𝑥 ↔ 𝑥 = 𝑧)
10		equcom 1932	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 ↔ 𝑧 = 𝑥)
11	9, 10	bitri 263	. . . . . . . . . 10 ⊢ (⟨𝑧, 𝑦⟩1st 𝑥 ↔ 𝑧 = 𝑥)
12	11	anbi1i 727	. . . . . . . . 9 ⊢ ((⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
13	5, 12	syl6bb 275	. . . . . . . 8 ⊢ (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
14	2, 13	ceqsexv 3215	. . . . . . 7 ⊢ (∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
15	14	exbii 1764	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
16		excom 2029	. . . . . 6 ⊢ (∃𝑧∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ ∃𝑝∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
17		opeq1 4340	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
18	17	eleq1d 2672	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
19	8, 18	ceqsexv 3215	. . . . . 6 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
20	15, 16, 19	3bitr3ri 290	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
21	20	exbii 1764	. . . 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 679	. . . . . . 7 ⊢ (((∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝 ∈ 𝐴) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
24	8	brres 5323	. . . . . . . . 9 ⊢ (𝑝(1st ↾ (V × V))𝑥 ↔ (𝑝1st 𝑥 ∧ 𝑝 ∈ (V × V)))
25		ancom 465	. . . . . . . . . 10 ⊢ ((𝑝1st 𝑥 ∧ 𝑝 ∈ (V × V)) ↔ (𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥))
26		elvv 5100	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (V × V) ↔ ∃𝑧∃𝑦 𝑝 = ⟨𝑧, 𝑦⟩)
27		excom 2029	. . . . . . . . . . . 12 ⊢ (∃𝑧∃𝑦 𝑝 = ⟨𝑧, 𝑦⟩ ↔ ∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
28	26, 27	bitri 263	. . . . . . . . . . 11 ⊢ (𝑝 ∈ (V × V) ↔ ∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
29	28	anbi1i 727	. . . . . . . . . 10 ⊢ ((𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
30	25, 29	bitri 263	. . . . . . . . 9 ⊢ ((𝑝1st 𝑥 ∧ 𝑝 ∈ (V × V)) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
31	24, 30	bitri 263	. . . . . . . 8 ⊢ (𝑝(1st ↾ (V × V))𝑥 ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
32	31	anbi1i 727	. . . . . . 7 ⊢ ((𝑝(1st ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴) ↔ ((∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝 ∈ 𝐴))
33		19.41vv 1902	. . . . . . 7 ⊢ (∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)) ↔ (∃𝑦∃𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
34	23, 32, 33	3bitr4i 291	. . . . . 6 ⊢ ((𝑝(1st ↾ (V × V))𝑥 ∧ 𝑝 ∈ 𝐴) ↔ ∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
35	22, 34	bitri 263	. . . . 5 ⊢ ((𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
36	35	exbii 1764	. . . 4 ⊢ (∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑝∃𝑦∃𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥 ∧ 𝑝 ∈ 𝐴)))
37	1, 21, 36	3bitr4i 291	. . 3 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥))
38	8	eldm2 5244	. . 3 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
39	8	elima2 5391	. . 3 ⊢ (𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴) ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ 𝑝(1st ↾ (V × V))𝑥))
40	37, 38, 39	3bitr4i 291	. 2 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴))
41	40	eqriv 2607	1 ⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)

Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   × cxp 5036  dom cdm 5038   ↾ cres 5040   “ cima 5041  1^st c1st 7057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059

This theorem is referenced by:  brdomain  31210