Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  oprabrexex2 Structured version   Visualization version   GIF version

Theorem oprabrexex2 7665
 Description: Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
Hypotheses
Ref Expression
oprabrexex2.1 𝐴 ∈ V
oprabrexex2.2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ∈ V
Assertion
Ref Expression
oprabrexex2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤𝐴 𝜑} ∈ V
Distinct variable group:   𝑥,𝐴,𝑦,𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem oprabrexex2
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 df-oprab 7144 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤𝐴 𝜑} = {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑)}
2 rexcom4 3237 . . . . 5 (∃𝑤𝐴𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑤𝐴𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
3 rexcom4 3237 . . . . . . 7 (∃𝑤𝐴𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦𝑤𝐴𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
4 rexcom4 3237 . . . . . . . . 9 (∃𝑤𝐴𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧𝑤𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
5 r19.42v 3331 . . . . . . . . . 10 (∃𝑤𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑))
65exbii 1849 . . . . . . . . 9 (∃𝑧𝑤𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑))
74, 6bitri 278 . . . . . . . 8 (∃𝑤𝐴𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑))
87exbii 1849 . . . . . . 7 (∃𝑦𝑤𝐴𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑))
93, 8bitri 278 . . . . . 6 (∃𝑤𝐴𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑))
109exbii 1849 . . . . 5 (∃𝑥𝑤𝐴𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑))
112, 10bitr2i 279 . . . 4 (∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑) ↔ ∃𝑤𝐴𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
1211abbii 2887 . . 3 {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑)} = {𝑣 ∣ ∃𝑤𝐴𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
131, 12eqtri 2845 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤𝐴 𝜑} = {𝑣 ∣ ∃𝑤𝐴𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
14 oprabrexex2.1 . . 3 𝐴 ∈ V
15 df-oprab 7144 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
16 oprabrexex2.2 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ∈ V
1715, 16eqeltrri 2911 . . 3 {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} ∈ V
1814, 17abrexex2 7656 . 2 {𝑣 ∣ ∃𝑤𝐴𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} ∈ V
1913, 18eqeltri 2910 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤𝐴 𝜑} ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2114  {cab 2800  ∃wrex 3131  Vcvv 3469  ⟨cop 4545  {coprab 7141 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446 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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-oprab 7144 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator