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Theorem dfdm5 33126
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 2166 . . . 4 (∃𝑦𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
2 opex 5321 . . . . . . . 8 𝑧, 𝑦⟩ ∈ V
3 breq1 5033 . . . . . . . . . 10 (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝1st 𝑥 ↔ ⟨𝑧, 𝑦⟩1st 𝑥))
4 eleq1 2877 . . . . . . . . . 10 (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝𝐴 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
53, 4anbi12d 633 . . . . . . . . 9 (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥𝑝𝐴) ↔ (⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
6 vex 3444 . . . . . . . . . . . 12 𝑧 ∈ V
7 vex 3444 . . . . . . . . . . . 12 𝑦 ∈ V
86, 7br1steq 33124 . . . . . . . . . . 11 (⟨𝑧, 𝑦⟩1st 𝑥𝑥 = 𝑧)
9 equcom 2025 . . . . . . . . . . 11 (𝑥 = 𝑧𝑧 = 𝑥)
108, 9bitri 278 . . . . . . . . . 10 (⟨𝑧, 𝑦⟩1st 𝑥𝑧 = 𝑥)
1110anbi1i 626 . . . . . . . . 9 ((⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
125, 11syl6bb 290 . . . . . . . 8 (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥𝑝𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
132, 12ceqsexv 3489 . . . . . . 7 (∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
1413exbii 1849 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
15 excom 2166 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ ∃𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
16 vex 3444 . . . . . . 7 𝑥 ∈ V
17 opeq1 4763 . . . . . . . 8 (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
1817eleq1d 2874 . . . . . . 7 (𝑧 = 𝑥 → (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
1916, 18ceqsexv 3489 . . . . . 6 (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
2014, 15, 193bitr3ri 305 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
2120exbii 1849 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
22 ancom 464 . . . . . 6 ((𝑝𝐴𝑝(1st ↾ (V × V))𝑥) ↔ (𝑝(1st ↾ (V × V))𝑥𝑝𝐴))
23 anass 472 . . . . . . 7 (((∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝𝐴) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
2416brresi 5827 . . . . . . . . 9 (𝑝(1st ↾ (V × V))𝑥 ↔ (𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥))
25 elvv 5590 . . . . . . . . . . 11 (𝑝 ∈ (V × V) ↔ ∃𝑧𝑦 𝑝 = ⟨𝑧, 𝑦⟩)
26 excom 2166 . . . . . . . . . . 11 (∃𝑧𝑦 𝑝 = ⟨𝑧, 𝑦⟩ ↔ ∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
2725, 26bitri 278 . . . . . . . . . 10 (𝑝 ∈ (V × V) ↔ ∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
2827anbi1i 626 . . . . . . . . 9 ((𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
2924, 28bitri 278 . . . . . . . 8 (𝑝(1st ↾ (V × V))𝑥 ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
3029anbi1i 626 . . . . . . 7 ((𝑝(1st ↾ (V × V))𝑥𝑝𝐴) ↔ ((∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝𝐴))
31 19.41vv 1951 . . . . . . 7 (∃𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
3223, 30, 313bitr4i 306 . . . . . 6 ((𝑝(1st ↾ (V × V))𝑥𝑝𝐴) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
3322, 32bitri 278 . . . . 5 ((𝑝𝐴𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
3433exbii 1849 . . . 4 (∃𝑝(𝑝𝐴𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
351, 21, 343bitr4i 306 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝(𝑝𝐴𝑝(1st ↾ (V × V))𝑥))
3616eldm2 5734 . . 3 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
3716elima2 5902 . . 3 (𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴) ↔ ∃𝑝(𝑝𝐴𝑝(1st ↾ (V × V))𝑥))
3835, 36, 373bitr4i 306 . 2 (𝑥 ∈ dom 𝐴𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴))
3938eqriv 2795 1 dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wex 1781  wcel 2111  Vcvv 3441  cop 4531   class class class wbr 5030   × cxp 5517  dom cdm 5519  cres 5521  cima 5522  1st c1st 7669
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-1st 7671
This theorem is referenced by:  brdomain  33504
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