Theorem elopabi 6435

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		relopab 4908	. . . 4 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		1st2nd 6419	. . . 4 ⊢ ((Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
3	1, 2	mpan 428	. . 3 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
4		id 19	. . 3 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5	3, 4	eqeltrrd 2316	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6		1stexg 6405	. . 3 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (1st ‘𝐴) ∈ V)
7		2ndexg 6406	. . 3 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (2nd ‘𝐴) ∈ V)
8		elopabi.1	. . . 4 ⊢ (𝑥 = (1st ‘𝐴) → (𝜑 ↔ 𝜓))
9		elopabi.2	. . . 4 ⊢ (𝑦 = (2nd ‘𝐴) → (𝜓 ↔ 𝜒))
10	8, 9	opelopabg 4412	. . 3 ⊢ (((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
11	6, 7, 10	syl2anc 415	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
12	5, 11	mpbid 147	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ⟨cop 3712  {copab 4192  Rel wrel 4781  ‘cfv 5380  1^st c1st 6376  2^nd c2nd 6377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fo 5386  df-fv 5388  df-1st 6378  df-2nd 6379

This theorem is used by:  exmidapne  7632  aprcl  8982  aptap  8986