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Theorem dfrn5 33074
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 2170 . . . 4 (∃𝑦𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
2 opex 5343 . . . . . . . 8 𝑦, 𝑧⟩ ∈ V
3 breq1 5055 . . . . . . . . . 10 (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝2nd 𝑥 ↔ ⟨𝑦, 𝑧⟩2nd 𝑥))
4 eleq1 2903 . . . . . . . . . 10 (𝑝 = ⟨𝑦, 𝑧⟩ → (𝑝𝐴 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
53, 4anbi12d 633 . . . . . . . . 9 (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥𝑝𝐴) ↔ (⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
6 vex 3483 . . . . . . . . . . . 12 𝑦 ∈ V
7 vex 3483 . . . . . . . . . . . 12 𝑧 ∈ V
86, 7br2ndeq 33072 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩2nd 𝑥𝑥 = 𝑧)
9 equcom 2026 . . . . . . . . . . 11 (𝑥 = 𝑧𝑧 = 𝑥)
108, 9bitri 278 . . . . . . . . . 10 (⟨𝑦, 𝑧⟩2nd 𝑥𝑧 = 𝑥)
1110anbi1i 626 . . . . . . . . 9 ((⟨𝑦, 𝑧⟩2nd 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
125, 11syl6bb 290 . . . . . . . 8 (𝑝 = ⟨𝑦, 𝑧⟩ → ((𝑝2nd 𝑥𝑝𝐴) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴)))
132, 12ceqsexv 3527 . . . . . . 7 (∃𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ (𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
1413exbii 1849 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ ∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴))
15 excom 2170 . . . . . 6 (∃𝑧𝑝(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ ∃𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
16 vex 3483 . . . . . . 7 𝑥 ∈ V
17 opeq2 4789 . . . . . . . 8 (𝑧 = 𝑥 → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑥⟩)
1817eleq1d 2900 . . . . . . 7 (𝑧 = 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴))
1916, 18ceqsexv 3527 . . . . . 6 (∃𝑧(𝑧 = 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
2014, 15, 193bitr3ri 305 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
2120exbii 1849 . . . 4 (∃𝑦𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑦𝑝𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
22 ancom 464 . . . . . 6 ((𝑝𝐴𝑝(2nd ↾ (V × V))𝑥) ↔ (𝑝(2nd ↾ (V × V))𝑥𝑝𝐴))
23 anass 472 . . . . . . 7 (((∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝𝐴) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
2416brresi 5849 . . . . . . . . 9 (𝑝(2nd ↾ (V × V))𝑥 ↔ (𝑝 ∈ (V × V) ∧ 𝑝2nd 𝑥))
25 elvv 5613 . . . . . . . . . 10 (𝑝 ∈ (V × V) ↔ ∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩)
2625anbi1i 626 . . . . . . . . 9 ((𝑝 ∈ (V × V) ∧ 𝑝2nd 𝑥) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥))
2724, 26bitri 278 . . . . . . . 8 (𝑝(2nd ↾ (V × V))𝑥 ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥))
2827anbi1i 626 . . . . . . 7 ((𝑝(2nd ↾ (V × V))𝑥𝑝𝐴) ↔ ((∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ 𝑝2nd 𝑥) ∧ 𝑝𝐴))
29 19.41vv 1952 . . . . . . 7 (∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)) ↔ (∃𝑦𝑧 𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
3023, 28, 293bitr4i 306 . . . . . 6 ((𝑝(2nd ↾ (V × V))𝑥𝑝𝐴) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
3122, 30bitri 278 . . . . 5 ((𝑝𝐴𝑝(2nd ↾ (V × V))𝑥) ↔ ∃𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
3231exbii 1849 . . . 4 (∃𝑝(𝑝𝐴𝑝(2nd ↾ (V × V))𝑥) ↔ ∃𝑝𝑦𝑧(𝑝 = ⟨𝑦, 𝑧⟩ ∧ (𝑝2nd 𝑥𝑝𝐴)))
331, 21, 323bitr4i 306 . . 3 (∃𝑦𝑦, 𝑥⟩ ∈ 𝐴 ↔ ∃𝑝(𝑝𝐴𝑝(2nd ↾ (V × V))𝑥))
3416elrn2 5808 . . 3 (𝑥 ∈ ran 𝐴 ↔ ∃𝑦𝑦, 𝑥⟩ ∈ 𝐴)
3516elima2 5922 . . 3 (𝑥 ∈ ((2nd ↾ (V × V)) “ 𝐴) ↔ ∃𝑝(𝑝𝐴𝑝(2nd ↾ (V × V))𝑥))
3633, 34, 353bitr4i 306 . 2 (𝑥 ∈ ran 𝐴𝑥 ∈ ((2nd ↾ (V × V)) “ 𝐴))
3736eqriv 2821 1 ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wex 1781  wcel 2115  Vcvv 3480  cop 4556   class class class wbr 5052   × cxp 5540  ran crn 5543  cres 5544  cima 5545  2nd c2nd 7683
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fo 6349  df-fv 6351  df-2nd 7685
This theorem is referenced by:  brrange  33452
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