Theorem elopabi 6174

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 4738	. . . 4 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		1st2nd 6160	. . . 4 ⊢ ((Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
3	1, 2	mpan 422	. . 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 2248	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6		1stexg 6146	. . 3 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (1st ‘𝐴) ∈ V)
7		2ndexg 6147	. . 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 4253	. . 3 ⊢ (((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
11	6, 7, 10	syl2anc 409	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))
12	5, 11	mpbid 146	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348   ∈ wcel 2141  Vcvv 2730  ⟨cop 3586  {copab 4049  Rel wrel 4616  ‘cfv 5198  1^st c1st 6117  2^nd c2nd 6118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206  df-1st 6119  df-2nd 6120

This theorem is referenced by:  aprcl  8565