Theorem elopabi 7875

Description: A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)

Hypotheses

Ref	Expression
elopabi.1	⊢ (𝑥 = (1st ‘𝐴) → (𝜑 ↔ 𝜓))
elopabi.2	⊢ (𝑦 = (2nd ‘𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
elopabi	⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)

Distinct variable groups:   𝑥,𝑦,𝐴   𝜒,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem elopabi

Step	Hyp	Ref	Expression
1		relopabv 5720	. . . 4 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		1st2nd 7853	. . . 4 ⊢ ((Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
3	1, 2	mpan 686	. . 3 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
4		id 22	. . 3 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5	3, 4	eqeltrrd 2840	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6		fvex 6769	. . 3 ⊢ (1st ‘𝐴) ∈ V
7		fvex 6769	. . 3 ⊢ (2nd ‘𝐴) ∈ V
8		elopabi.1	. . 3 ⊢ (𝑥 = (1st ‘𝐴) → (𝜑 ↔ 𝜓))
9		elopabi.2	. . 3 ⊢ (𝑦 = (2nd ‘𝐴) → (𝜓 ↔ 𝜒))
10	6, 7, 8, 9	opelopab 5448	. 2 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
11	5, 10	sylib 217	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  ⟨cop 4564  {copab 5132  Rel wrel 5585  ‘cfv 6418  1^st c1st 7802  2^nd c2nd 7803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-1st 7804  df-2nd 7805

This theorem is referenced by:  vciOLD  28824  sat1el2xp  33241  drngoi  36036  dicelval1sta  39128