Theorem elopabi 8061

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 5800	. . . 4 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		1st2nd 8038	. . . 4 ⊢ ((Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
3	1, 2	mpan 690	. . 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 2835	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6		fvex 6889	. . 3 ⊢ (1st ‘𝐴) ∈ V
7		fvex 6889	. . 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 5517	. 2 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
11	5, 10	sylib 218	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  ⟨cop 4607  {copab 5181  Rel wrel 5659  ‘cfv 6531  1^st c1st 7986  2^nd c2nd 7987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fv 6539  df-1st 7988  df-2nd 7989

This theorem is referenced by:  vciOLD  30542  sat1el2xp  35401  drngoi  37975  dicelval1sta  41206