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Theorem elopabi 8055

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 = (1^st ‘𝐴) → (𝜑 ↔ 𝜓))
elopabi.2	⊢ (𝑦 = (2^nd ‘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 5808	. . . 4 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		1st2nd 8032	. . . 4 ⊢ ((Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
3	1, 2	mpan 702	. . 3 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
4		id 23	. . 3 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5	3, 4	eqeltrrd 2864	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6		fvex 6894	. . 3 ⊢ (1^st ‘𝐴) ∈ V
7		fvex 6894	. . 3 ⊢ (2^nd ‘𝐴) ∈ V
8		elopabi.1	. . 3 ⊢ (𝑥 = (1^st ‘𝐴) → (𝜑 ↔ 𝜓))
9		elopabi.2	. . 3 ⊢ (𝑦 = (2^nd ‘𝐴) → (𝜓 ↔ 𝜒))
10	6, 7, 8, 9	opelopab 5527	. 2 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
11	5, 10	sylib 221	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 = wceq 1570 ∈ wcel 2143 ⟨cop 4595 {copab 5173 Rel wrel 5666 ‘cfv 6536 1^st c1st 7980 2^nd c2nd 7981

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pr 5404 ax-un 7732

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-iota 6492 df-fun 6538 df-fv 6544 df-1st 7982 df-2nd 7983

This theorem is used by: vciOLD 30922 sat1el2xp 35879 drngoi 38630 dicelval1sta 41989

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