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Theorem dfrn5 31376
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 2039 . . . 4 (∃𝑦𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
2 opex 4893 . . . . . . . 8 𝑦, 𝑧⟩ ∈ V
3 breq1 4616 . . . . . . . . . 10 (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝2nd 𝑥 ↔ ⟨𝑦, 𝑧⟩2nd 𝑥))
4 eleq1 2686 . . . . . . . . . 10 (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
53, 4anbi12d 746 . . . . . . . . 9 (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥𝑝𝐴) ↔ (⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
6 vex 3189 . . . . . . . . . . . 12 𝑦 ∈ V
7 vex 3189 . . . . . . . . . . . 12 𝑧 ∈ V
8 vex 3189 . . . . . . . . . . . 12 𝑥 ∈ V
96, 7, 8br2ndeq 31372 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩2nd 𝑥𝑥 = 𝑧)
10 equcom 1942 . . . . . . . . . . 11 (𝑥 = 𝑧𝑧 = 𝑥)
119, 10bitri 264 . . . . . . . . . 10 (⟨𝑦, 𝑧⟩2nd 𝑥𝑧 = 𝑥)
1211anbi1i 730 . . . . . . . . 9 ((⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
135, 12syl6bb 276 . . . . . . . 8 (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥𝑝𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
142, 13ceqsexv 3228 . . . . . . 7 (∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
1514exbii 1771 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
16 excom 2039 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ ∃𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
17 opeq2 4371 . . . . . . . 8 (𝑧 = 𝑥 → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑥⟩)
1817eleq1d 2683 . . . . . . 7 (𝑧 = 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴))
198, 18ceqsexv 3228 . . . . . 6 (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
2015, 16, 193bitr3ri 291 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
2120exbii 1771 . . . 4 (∃𝑦𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑦𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
22 ancom 466 . . . . . 6 ((𝑝𝐴𝑝(2nd ↾ (V × V))𝑥) ↔ (𝑝(2nd ↾ (V × V))𝑥𝑝𝐴))
23 anass 680 . . . . . . 7 (((∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝𝐴) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
248brres 5362 . . . . . . . . 9 (𝑝(2nd ↾ (V × V))𝑥 ↔ (𝑝2nd 𝑥𝑝 ∈ (V × V)))
25 ancom 466 . . . . . . . . . 10 ((𝑝2nd 𝑥𝑝 ∈ (V × V)) ↔ (𝑝 ∈ (V × V) ∧ 𝑝2nd 𝑥))
26 elvv 5138 . . . . . . . . . . 11 (𝑝 ∈ (V × V) ↔ ∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩)
2726anbi1i 730 . . . . . . . . . 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 730 . . . . . . 7 ((𝑝(2nd ↾ (V × V))𝑥𝑝𝐴) ↔ ((∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝𝐴))
31 19.41vv 1912 . . . . . . 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 1771 . . . 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))
368elrn2 5325 . . 3 (𝑥 ∈ ran 𝐴 ↔ ∃𝑦𝑦, 𝑥⟩ ∈ 𝐴)
378elima2 5431 . . 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 2618 1 ran 𝐴 = ((2nd ↾ (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  ran crn 5075  cres 5076  cima 5077  2nd c2nd 7112
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-2nd 7114
This theorem is referenced by:  brrange  31680
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