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Theorem dfdm5 31375
Description: Definition of domain in terms of 1st 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.
StepHypRef Expression
1 excom 2039 . . . 4 (∃𝑦𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
2 opex 4893 . . . . . . . 8 𝑧, 𝑦⟩ ∈ V
3 breq1 4616 . . . . . . . . . 10 (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝1st 𝑥 ↔ ⟨𝑧, 𝑦⟩1st 𝑥))
4 eleq1 2686 . . . . . . . . . 10 (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝𝐴 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
53, 4anbi12d 746 . . . . . . . . 9 (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥𝑝𝐴) ↔ (⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
6 vex 3189 . . . . . . . . . . . 12 𝑧 ∈ V
7 vex 3189 . . . . . . . . . . . 12 𝑦 ∈ V
8 vex 3189 . . . . . . . . . . . 12 𝑥 ∈ V
96, 7, 8br1steq 31371 . . . . . . . . . . 11 (⟨𝑧, 𝑦⟩1st 𝑥𝑥 = 𝑧)
10 equcom 1942 . . . . . . . . . . 11 (𝑥 = 𝑧𝑧 = 𝑥)
119, 10bitri 264 . . . . . . . . . 10 (⟨𝑧, 𝑦⟩1st 𝑥𝑧 = 𝑥)
1211anbi1i 730 . . . . . . . . 9 ((⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
135, 12syl6bb 276 . . . . . . . 8 (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥𝑝𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
142, 13ceqsexv 3228 . . . . . . 7 (∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
1514exbii 1771 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
16 excom 2039 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ ∃𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
17 opeq1 4370 . . . . . . . 8 (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
1817eleq1d 2683 . . . . . . 7 (𝑧 = 𝑥 → (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
198, 18ceqsexv 3228 . . . . . 6 (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
2015, 16, 193bitr3ri 291 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
2120exbii 1771 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
22 ancom 466 . . . . . 6 ((𝑝𝐴𝑝(1st ↾ (V × V))𝑥) ↔ (𝑝(1st ↾ (V × V))𝑥𝑝𝐴))
23 anass 680 . . . . . . 7 (((∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝𝐴) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
248brres 5362 . . . . . . . . 9 (𝑝(1st ↾ (V × V))𝑥 ↔ (𝑝1st 𝑥𝑝 ∈ (V × V)))
25 ancom 466 . . . . . . . . . 10 ((𝑝1st 𝑥𝑝 ∈ (V × V)) ↔ (𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥))
26 elvv 5138 . . . . . . . . . . . 12 (𝑝 ∈ (V × V) ↔ ∃𝑧𝑦 𝑝 = ⟨𝑧, 𝑦⟩)
27 excom 2039 . . . . . . . . . . . 12 (∃𝑧𝑦 𝑝 = ⟨𝑧, 𝑦⟩ ↔ ∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
2826, 27bitri 264 . . . . . . . . . . 11 (𝑝 ∈ (V × V) ↔ ∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
2928anbi1i 730 . . . . . . . . . 10 ((𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
3025, 29bitri 264 . . . . . . . . 9 ((𝑝1st 𝑥𝑝 ∈ (V × V)) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
3124, 30bitri 264 . . . . . . . 8 (𝑝(1st ↾ (V × V))𝑥 ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
3231anbi1i 730 . . . . . . 7 ((𝑝(1st ↾ (V × V))𝑥𝑝𝐴) ↔ ((∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝𝐴))
33 19.41vv 1912 . . . . . . 7 (∃𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
3423, 32, 333bitr4i 292 . . . . . 6 ((𝑝(1st ↾ (V × V))𝑥𝑝𝐴) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
3522, 34bitri 264 . . . . 5 ((𝑝𝐴𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
3635exbii 1771 . . . 4 (∃𝑝(𝑝𝐴𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
371, 21, 363bitr4i 292 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝(𝑝𝐴𝑝(1st ↾ (V × V))𝑥))
388eldm2 5282 . . 3 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
398elima2 5431 . . 3 (𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴) ↔ ∃𝑝(𝑝𝐴𝑝(1st ↾ (V × V))𝑥))
4037, 38, 393bitr4i 292 . 2 (𝑥 ∈ dom 𝐴𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴))
4140eqriv 2618 1 dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  cop 4154   class class class wbr 4613   × cxp 5072  dom cdm 5074  cres 5076  cima 5077  1st c1st 7111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-1st 7113
This theorem is referenced by:  brdomain  31679
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