1stval2 - Metamath Proof Explorer

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Theorem 1stval2 8003

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 5730	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3454	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op1st 7994	. . . . 5 ⊢ (1^st ‘⟨𝑥, 𝑦⟩) = 𝑥
5	2, 3	op1stb 5447	. . . . 5 ⊢ ∩ ∩ ⟨𝑥, 𝑦⟩ = 𝑥
6	4, 5	eqtr4i 2786	. . . 4 ⊢ (1^st ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ⟨𝑥, 𝑦⟩
7		fveq2 6878	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = (1^st ‘⟨𝑥, 𝑦⟩))
8		inteq 4910	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ 𝐴 = ∩ ⟨𝑥, 𝑦⟩)
9	8	inteqd 4912	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐴 = ∩ ∩ ⟨𝑥, 𝑦⟩)
10	6, 7, 9	3eqtr4a 2821	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = ∩ ∩ 𝐴)
11	10	exlimivv 1965	. 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

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∃wex 1812 ∈ wcel 2145 Vcvv 3450 ⟨cop 4590 ∩ cint 4907 × cxp 5653 ‘cfv 6533 1^st c1st 7984

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pr 5398 ax-un 7736

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 2100 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-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-iota 6489 df-fun 6535 df-fv 6541 df-1st 7986

This theorem is used by: 1stdm 8037

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