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Theorem dfrn5 35500
Description: Definition of range in terms of 2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
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 2151 . . . 4 (∃𝑦𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
2 opex 5466 . . . . . . . 8 𝑦, 𝑧⟩ ∈ V
3 breq1 5152 . . . . . . . . . 10 (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝2nd 𝑥 ↔ ⟨𝑦, 𝑧⟩2nd 𝑥))
4 eleq1 2813 . . . . . . . . . 10 (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
53, 4anbi12d 630 . . . . . . . . 9 (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥𝑝𝐴) ↔ (⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
6 vex 3465 . . . . . . . . . . . 12 𝑦 ∈ V
7 vex 3465 . . . . . . . . . . . 12 𝑧 ∈ V
86, 7br2ndeq 35498 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩2nd 𝑥𝑥 = 𝑧)
9 equcom 2013 . . . . . . . . . . 11 (𝑥 = 𝑧𝑧 = 𝑥)
108, 9bitri 274 . . . . . . . . . 10 (⟨𝑦, 𝑧⟩2nd 𝑥𝑧 = 𝑥)
1110anbi1i 622 . . . . . . . . 9 ((⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
125, 11bitrdi 286 . . . . . . . 8 (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥𝑝𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
132, 12ceqsexv 3514 . . . . . . 7 (∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
1413exbii 1842 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
15 excom 2151 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ ∃𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
16 vex 3465 . . . . . . 7 𝑥 ∈ V
17 opeq2 4876 . . . . . . . 8 (𝑧 = 𝑥 → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑥⟩)
1817eleq1d 2810 . . . . . . 7 (𝑧 = 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴))
1916, 18ceqsexv 3514 . . . . . 6 (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
2014, 15, 193bitr3ri 301 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
2120exbii 1842 . . . 4 (∃𝑦𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑦𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
22 ancom 459 . . . . . 6 ((𝑝𝐴𝑝(2nd ↾ (V × V))𝑥) ↔ (𝑝(2nd ↾ (V × V))𝑥𝑝𝐴))
23 anass 467 . . . . . . 7 (((∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝𝐴) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
2416brresi 5994 . . . . . . . . 9 (𝑝(2nd ↾ (V × V))𝑥 ↔ (𝑝 ∈ (V × V) ∧ 𝑝2nd 𝑥))
25 elvv 5752 . . . . . . . . . 10 (𝑝 ∈ (V × V) ↔ ∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩)
2625anbi1i 622 . . . . . . . . 9 ((𝑝 ∈ (V × V) ∧ 𝑝2nd 𝑥) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥))
2724, 26bitri 274 . . . . . . . 8 (𝑝(2nd ↾ (V × V))𝑥 ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥))
2827anbi1i 622 . . . . . . 7 ((𝑝(2nd ↾ (V × V))𝑥𝑝𝐴) ↔ ((∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝𝐴))
29 19.41vv 1946 . . . . . . 7 (∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
3023, 28, 293bitr4i 302 . . . . . 6 ((𝑝(2nd ↾ (V × V))𝑥𝑝𝐴) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
3122, 30bitri 274 . . . . 5 ((𝑝𝐴𝑝(2nd ↾ (V × V))𝑥) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
3231exbii 1842 . . . 4 (∃𝑝(𝑝𝐴𝑝(2nd ↾ (V × V))𝑥) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
331, 21, 323bitr4i 302 . . 3 (∃𝑦𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑝(𝑝𝐴𝑝(2nd ↾ (V × V))𝑥))
3416elrn2 5895 . . 3 (𝑥 ∈ ran 𝐴 ↔ ∃𝑦𝑦, 𝑥⟩ ∈ 𝐴)
3516elima2 6070 . . 3 (𝑥 ∈ ((2nd ↾ (V × V)) “ 𝐴) ↔ ∃𝑝(𝑝𝐴𝑝(2nd ↾ (V × V))𝑥))
3633, 34, 353bitr4i 302 . 2 (𝑥 ∈ ran 𝐴𝑥 ∈ ((2nd ↾ (V × V)) “ 𝐴))
3736eqriv 2722 1 ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wex 1773  wcel 2098  Vcvv 3461  cop 4636   class class class wbr 5149   × cxp 5676  ran crn 5679  cres 5680  cima 5681  2nd c2nd 7993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fo 6555  df-fv 6557  df-2nd 7995
This theorem is referenced by:  brrange  35661
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