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Theorem elopabi 8095

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 5804	. . . 4 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		1st2nd 8071	. . . 4 ⊢ ((Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
3	1, 2	mpan 703	. . 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 2865	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6		fvex 6918	. . 3 ⊢ (1^st ‘𝐴) ∈ V
7		fvex 6918	. . 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 5521	. 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 2147 ⟨cop 4838 {copab 5170 Rel wrel 5660 ‘cfv 6555 1^st c1st 8020 2^nd c2nd 8021

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pr 5395 ax-un 7772

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-iota 6509 df-fun 6557 df-fv 6563 df-1st 8022 df-2nd 8023

This theorem is used by: vciOLD 31190 sat1el2xp 36310 drngoi 39076 dicelval1sta 42435

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