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Theorem dfdm5 32573
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 2099 . . . 4 (∃𝑦𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
2 opex 5209 . . . . . . . 8 𝑧, 𝑦⟩ ∈ V
3 breq1 4928 . . . . . . . . . 10 (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝1st 𝑥 ↔ ⟨𝑧, 𝑦⟩1st 𝑥))
4 eleq1 2846 . . . . . . . . . 10 (𝑝 = ⟨𝑧, 𝑦⟩ → (𝑝𝐴 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
53, 4anbi12d 622 . . . . . . . . 9 (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥𝑝𝐴) ↔ (⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
6 vex 3411 . . . . . . . . . . . 12 𝑧 ∈ V
7 vex 3411 . . . . . . . . . . . 12 𝑦 ∈ V
86, 7br1steq 32571 . . . . . . . . . . 11 (⟨𝑧, 𝑦⟩1st 𝑥𝑥 = 𝑧)
9 equcom 1976 . . . . . . . . . . 11 (𝑥 = 𝑧𝑧 = 𝑥)
108, 9bitri 267 . . . . . . . . . 10 (⟨𝑧, 𝑦⟩1st 𝑥𝑧 = 𝑥)
1110anbi1i 615 . . . . . . . . 9 ((⟨𝑧, 𝑦⟩1st 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
125, 11syl6bb 279 . . . . . . . 8 (𝑝 = ⟨𝑧, 𝑦⟩ → ((𝑝1st 𝑥𝑝𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)))
132, 12ceqsexv 3455 . . . . . . 7 (∃𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
1413exbii 1811 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴))
15 excom 2099 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ ∃𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
16 vex 3411 . . . . . . 7 𝑥 ∈ V
17 opeq1 4673 . . . . . . . 8 (𝑧 = 𝑥 → ⟨𝑧, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
1817eleq1d 2843 . . . . . . 7 (𝑧 = 𝑥 → (⟨𝑧, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
1916, 18ceqsexv 3455 . . . . . 6 (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
2014, 15, 193bitr3ri 294 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
2120exbii 1811 . . . 4 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑦𝑝𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
22 ancom 453 . . . . . 6 ((𝑝𝐴𝑝(1st ↾ (V × V))𝑥) ↔ (𝑝(1st ↾ (V × V))𝑥𝑝𝐴))
23 anass 461 . . . . . . 7 (((∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝𝐴) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
2416brresi 5701 . . . . . . . . 9 (𝑝(1st ↾ (V × V))𝑥 ↔ (𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥))
25 elvv 5472 . . . . . . . . . . 11 (𝑝 ∈ (V × V) ↔ ∃𝑧𝑦 𝑝 = ⟨𝑧, 𝑦⟩)
26 excom 2099 . . . . . . . . . . 11 (∃𝑧𝑦 𝑝 = ⟨𝑧, 𝑦⟩ ↔ ∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
2725, 26bitri 267 . . . . . . . . . 10 (𝑝 ∈ (V × V) ↔ ∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩)
2827anbi1i 615 . . . . . . . . 9 ((𝑝 ∈ (V × V) ∧ 𝑝1st 𝑥) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
2924, 28bitri 267 . . . . . . . 8 (𝑝(1st ↾ (V × V))𝑥 ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥))
3029anbi1i 615 . . . . . . 7 ((𝑝(1st ↾ (V × V))𝑥𝑝𝐴) ↔ ((∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ 𝑝1st 𝑥) ∧ 𝑝𝐴))
31 19.41vv 1910 . . . . . . 7 (∃𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
3223, 30, 313bitr4i 295 . . . . . 6 ((𝑝(1st ↾ (V × V))𝑥𝑝𝐴) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
3322, 32bitri 267 . . . . 5 ((𝑝𝐴𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
3433exbii 1811 . . . 4 (∃𝑝(𝑝𝐴𝑝(1st ↾ (V × V))𝑥) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑧, 𝑦⟩ ∧ (𝑝1st 𝑥𝑝𝐴)))
351, 21, 343bitr4i 295 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑝(𝑝𝐴𝑝(1st ↾ (V × V))𝑥))
3616eldm2 5616 . . 3 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
3716elima2 5773 . . 3 (𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴) ↔ ∃𝑝(𝑝𝐴𝑝(1st ↾ (V × V))𝑥))
3835, 36, 373bitr4i 295 . 2 (𝑥 ∈ dom 𝐴𝑥 ∈ ((1st ↾ (V × V)) “ 𝐴))
3938eqriv 2768 1 dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 387   = wceq 1508  wex 1743  wcel 2051  Vcvv 3408  cop 4441   class class class wbr 4925   × cxp 5401  dom cdm 5403  cres 5405  cima 5406  1st c1st 7497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-fo 6191  df-fv 6193  df-1st 7499
This theorem is referenced by:  brdomain  32952
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