Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfrn5 Structured version   Visualization version   GIF version

Theorem dfrn5 31801
Description: Definition of range in terms of 2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
dfrn5 ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)

Proof of Theorem dfrn5
Dummy variables 𝑝 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 excom 2082 . . . 4 (∃𝑦𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
2 opex 4962 . . . . . . . 8 𝑦, 𝑧⟩ ∈ V
3 breq1 4688 . . . . . . . . . 10 (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝2nd 𝑥 ↔ ⟨𝑦, 𝑧⟩2nd 𝑥))
4 eleq1 2718 . . . . . . . . . 10 (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
53, 4anbi12d 747 . . . . . . . . 9 (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥𝑝𝐴) ↔ (⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
6 vex 3234 . . . . . . . . . . . 12 𝑦 ∈ V
7 vex 3234 . . . . . . . . . . . 12 𝑧 ∈ V
86, 7br2ndeq 31797 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩2nd 𝑥𝑥 = 𝑧)
9 equcom 1991 . . . . . . . . . . 11 (𝑥 = 𝑧𝑧 = 𝑥)
108, 9bitri 264 . . . . . . . . . 10 (⟨𝑦, 𝑧⟩2nd 𝑥𝑧 = 𝑥)
1110anbi1i 731 . . . . . . . . 9 ((⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
125, 11syl6bb 276 . . . . . . . 8 (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥𝑝𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
132, 12ceqsexv 3273 . . . . . . 7 (∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
1413exbii 1814 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
15 excom 2082 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ ∃𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
16 vex 3234 . . . . . . 7 𝑥 ∈ V
17 opeq2 4434 . . . . . . . 8 (𝑧 = 𝑥 → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑥⟩)
1817eleq1d 2715 . . . . . . 7 (𝑧 = 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴))
1916, 18ceqsexv 3273 . . . . . 6 (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
2014, 15, 193bitr3ri 291 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
2120exbii 1814 . . . 4 (∃𝑦𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑦𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
22 ancom 465 . . . . . 6 ((𝑝𝐴𝑝(2nd ↾ (V × V))𝑥) ↔ (𝑝(2nd ↾ (V × V))𝑥𝑝𝐴))
23 anass 682 . . . . . . 7 (((∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝𝐴) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
2416brres 5437 . . . . . . . . 9 (𝑝(2nd ↾ (V × V))𝑥 ↔ (𝑝2nd 𝑥𝑝 ∈ (V × V)))
25 ancom 465 . . . . . . . . . 10 ((𝑝2nd 𝑥𝑝 ∈ (V × V)) ↔ (𝑝 ∈ (V × V) ∧ 𝑝2nd 𝑥))
26 elvv 5211 . . . . . . . . . . 11 (𝑝 ∈ (V × V) ↔ ∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩)
2726anbi1i 731 . . . . . . . . . 10 ((𝑝 ∈ (V × V) ∧ 𝑝2nd 𝑥) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥))
2825, 27bitri 264 . . . . . . . . 9 ((𝑝2nd 𝑥𝑝 ∈ (V × V)) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥))
2924, 28bitri 264 . . . . . . . 8 (𝑝(2nd ↾ (V × V))𝑥 ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥))
3029anbi1i 731 . . . . . . 7 ((𝑝(2nd ↾ (V × V))𝑥𝑝𝐴) ↔ ((∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝𝐴))
31 19.41vv 1918 . . . . . . 7 (∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
3223, 30, 313bitr4i 292 . . . . . 6 ((𝑝(2nd ↾ (V × V))𝑥𝑝𝐴) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
3322, 32bitri 264 . . . . 5 ((𝑝𝐴𝑝(2nd ↾ (V × V))𝑥) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
3433exbii 1814 . . . 4 (∃𝑝(𝑝𝐴𝑝(2nd ↾ (V × V))𝑥) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
351, 21, 343bitr4i 292 . . 3 (∃𝑦𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑝(𝑝𝐴𝑝(2nd ↾ (V × V))𝑥))
3616elrn2 5397 . . 3 (𝑥 ∈ ran 𝐴 ↔ ∃𝑦𝑦, 𝑥⟩ ∈ 𝐴)
3716elima2 5507 . . 3 (𝑥 ∈ ((2nd ↾ (V × V)) “ 𝐴) ↔ ∃𝑝(𝑝𝐴𝑝(2nd ↾ (V × V))𝑥))
3835, 36, 373bitr4i 292 . 2 (𝑥 ∈ ran 𝐴𝑥 ∈ ((2nd ↾ (V × V)) “ 𝐴))
3938eqriv 2648 1 ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cop 4216   class class class wbr 4685   × cxp 5141  ran crn 5144  cres 5145  cima 5146  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-2nd 7211
This theorem is referenced by:  brrange  32166
  Copyright terms: Public domain W3C validator