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Theorem oprabrexex2 7355
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 6845 . . 3 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤𝐴 𝜑} = {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑)}
2 rexcom4 3377 . . . . 5 (∃𝑤𝐴𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑤𝐴𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
3 rexcom4 3377 . . . . . . 7 (∃𝑤𝐴𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦𝑤𝐴𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
4 rexcom4 3377 . . . . . . . . 9 (∃𝑤𝐴𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧𝑤𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
5 r19.42v 3238 . . . . . . . . . 10 (∃𝑤𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑))
65exbii 1943 . . . . . . . . 9 (∃𝑧𝑤𝐴 (𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑))
74, 6bitri 266 . . . . . . . 8 (∃𝑤𝐴𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑))
87exbii 1943 . . . . . . 7 (∃𝑦𝑤𝐴𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑))
93, 8bitri 266 . . . . . 6 (∃𝑤𝐴𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑))
109exbii 1943 . . . . 5 (∃𝑥𝑤𝐴𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑))
112, 10bitr2i 267 . . . 4 (∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑) ↔ ∃𝑤𝐴𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
1211abbii 2881 . . 3 {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ ∃𝑤𝐴 𝜑)} = {𝑣 ∣ ∃𝑤𝐴𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
131, 12eqtri 2786 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤𝐴 𝜑} = {𝑣 ∣ ∃𝑤𝐴𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
14 oprabrexex2.1 . . 3 𝐴 ∈ V
15 df-oprab 6845 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
16 oprabrexex2.2 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ∈ V
1715, 16eqeltrri 2840 . . 3 {𝑣 ∣ ∃𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} ∈ V
1814, 17abrexex2 7345 . 2 {𝑣 ∣ ∃𝑤𝐴𝑥𝑦𝑧(𝑣 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} ∈ V
1913, 18eqeltri 2839 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑤𝐴 𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wex 1874  wcel 2155  {cab 2750  wrex 3055  Vcvv 3349  cop 4339  {coprab 6842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pr 5061  ax-un 7146
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-oprab 6845
This theorem is referenced by: (None)
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