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Theorem dfdm5 34386
Description: Definition of domain in terms of 1st 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.
StepHypRef Expression
1 excom 2163 . . . 4 (∃𝑦𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
2 opex 5426 . . . . . . . 8 𝑧, 𝑦⟩ ∈ V
3 breq1 5113 . . . . . . . . . 10 (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝1st 𝑥 ↔ ⟨𝑧, 𝑦⟩1st 𝑥))
4 eleq1 2826 . . . . . . . . . 10 (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝𝐴 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
53, 4anbi12d 632 . . . . . . . . 9 (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥𝑝𝐴) ↔ (⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
6 vex 3452 . . . . . . . . . . . 12 𝑧 ∈ V
7 vex 3452 . . . . . . . . . . . 12 𝑦 ∈ V
86, 7br1steq 34384 . . . . . . . . . . 11 (⟨𝑧, 𝑦⟩1st 𝑥𝑥 = 𝑧)
9 equcom 2022 . . . . . . . . . . 11 (𝑥 = 𝑧𝑧 = 𝑥)
108, 9bitri 275 . . . . . . . . . 10 (⟨𝑧, 𝑦⟩1st 𝑥𝑧 = 𝑥)
1110anbi1i 625 . . . . . . . . 9 ((⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
125, 11bitrdi 287 . . . . . . . 8 (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥𝑝𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
132, 12ceqsexv 3497 . . . . . . 7 (∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
1413exbii 1851 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
15 excom 2163 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ ∃𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
16 vex 3452 . . . . . . 7 𝑥 ∈ V
17 opeq1 4835 . . . . . . . 8 (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
1817eleq1d 2823 . . . . . . 7 (𝑧 = 𝑥 → (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
1916, 18ceqsexv 3497 . . . . . 6 (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
2014, 15, 193bitr3ri 302 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
2120exbii 1851 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
22 ancom 462 . . . . . 6 ((𝑝𝐴𝑝(1st ↾ (V × V))𝑥) ↔ (𝑝(1st ↾ (V × V))𝑥𝑝𝐴))
23 anass 470 . . . . . . 7 (((∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝𝐴) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
2416brresi 5951 . . . . . . . . 9 (𝑝(1st ↾ (V × V))𝑥 ↔ (𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥))
25 elvv 5711 . . . . . . . . . . 11 (𝑝 ∈ (V × V) ↔ ∃𝑧𝑦 𝑝 = ⟨𝑧, 𝑦⟩)
26 excom 2163 . . . . . . . . . . 11 (∃𝑧𝑦 𝑝 = ⟨𝑧, 𝑦⟩ ↔ ∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
2725, 26bitri 275 . . . . . . . . . 10 (𝑝 ∈ (V × V) ↔ ∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
2827anbi1i 625 . . . . . . . . 9 ((𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
2924, 28bitri 275 . . . . . . . 8 (𝑝(1st ↾ (V × V))𝑥 ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
3029anbi1i 625 . . . . . . 7 ((𝑝(1st ↾ (V × V))𝑥𝑝𝐴) ↔ ((∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝𝐴))
31 19.41vv 1955 . . . . . . 7 (∃𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
3223, 30, 313bitr4i 303 . . . . . 6 ((𝑝(1st ↾ (V × V))𝑥𝑝𝐴) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
3322, 32bitri 275 . . . . 5 ((𝑝𝐴𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
3433exbii 1851 . . . 4 (∃𝑝(𝑝𝐴𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
351, 21, 343bitr4i 303 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝(𝑝𝐴𝑝(1st ↾ (V × V))𝑥))
3616eldm2 5862 . . 3 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
3716elima2 6024 . . 3 (𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴) ↔ ∃𝑝(𝑝𝐴𝑝(1st ↾ (V × V))𝑥))
3835, 36, 373bitr4i 303 . 2 (𝑥 ∈ dom 𝐴𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴))
3938eqriv 2734 1 dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3448  cop 4597   class class class wbr 5110   × cxp 5636  dom cdm 5638  cres 5640  cima 5641  1st c1st 7924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fo 6507  df-fv 6509  df-1st 7926
This theorem is referenced by:  brdomain  34547
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