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Theorem 1stval2 7724

Description: Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)

**Assertion**
Ref	Expression
1stval2	⊢ (𝐴 ∈ (V × V) → (1^st ‘𝐴) = ∩ ∩ 𝐴)

Proof of Theorem 1stval2 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elvv 5591	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 3401	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3401	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op1st 7715	. . . . 5 ⊢ (1^st ‘⟨𝑥, 𝑦⟩) = 𝑥
5	2, 3	op1stb 5326	. . . . 5 ⊢ ∩ ∩ ⟨𝑥, 𝑦⟩ = 𝑥
6	4, 5	eqtr4i 2764	. . . 4 ⊢ (1^st ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ⟨𝑥, 𝑦⟩
7		fveq2 6668	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = (1^st ‘⟨𝑥, 𝑦⟩))
8		inteq 4836	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ 𝐴 = ∩ ⟨𝑥, 𝑦⟩)
9	8	inteqd 4838	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐴 = ∩ ∩ ⟨𝑥, 𝑦⟩)
10	6, 7, 9	3eqtr4a 2799	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = ∩ ∩ 𝐴)
11	10	exlimivv 1938	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = ∩ ∩ 𝐴)
12	1, 11	sylbi 220	1 ⊢ (𝐴 ∈ (V × V) → (1^st ‘𝐴) = ∩ ∩ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∃wex 1786 ∈ wcel 2113 Vcvv 3397 ⟨cop 4519 ∩ cint 4833 × cxp 5517 ‘cfv 6333 1^st c1st 7705

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 ax-un 7473

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-int 4834 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6291 df-fun 6335 df-fv 6341 df-1st 7707

This theorem is referenced by: 1stdm 7757

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