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Theorem elopabi 8073

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 5798	. . . 4 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2		1st2nd 8050	. . . 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 2861	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6		fvex 6898	. . 3 ⊢ (1^st ‘𝐴) ∈ V
7		fvex 6898	. . 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 5516	. 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 1569 ∈ wcel 2144 ⟨cop 4589 {copab 5166 Rel wrel 5655 ‘cfv 6538 1^st c1st 7999 2^nd c2nd 8000

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-iota 6494 df-fun 6540 df-fv 6546 df-1st 8001 df-2nd 8002

This theorem is used by: vciOLD 31163 sat1el2xp 36144 drngoi 38886 dicelval1sta 42245

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