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Theorem 1stval2 7338

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 5322	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 3331	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3331	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op1st 7329	. . . . 5 ⊢ (1^st ‘⟨𝑥, 𝑦⟩) = 𝑥
5	2, 3	op1stb 5076	. . . . 5 ⊢ ∩ ∩ ⟨𝑥, 𝑦⟩ = 𝑥
6	4, 5	eqtr4i 2773	. . . 4 ⊢ (1^st ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ⟨𝑥, 𝑦⟩
7		fveq2 6340	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = (1^st ‘⟨𝑥, 𝑦⟩))
8		inteq 4618	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ 𝐴 = ∩ ⟨𝑥, 𝑦⟩)
9	8	inteqd 4620	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐴 = ∩ ∩ ⟨𝑥, 𝑦⟩)
10	6, 7, 9	3eqtr4a 2808	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = ∩ ∩ 𝐴)
11	10	exlimivv 1997	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝐴) = ∩ ∩ 𝐴)
12	1, 11	sylbi 207	1 ⊢ (𝐴 ∈ (V × V) → (1^st ‘𝐴) = ∩ ∩ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1620 ∃wex 1841 ∈ wcel 2127 Vcvv 3328 ⟨cop 4315 ∩ cint 4615 × cxp 5252 ‘cfv 6037 1^st c1st 7319

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ral 3043 df-rex 3044 df-rab 3047 df-v 3330 df-sbc 3565 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-sn 4310 df-pr 4312 df-op 4316 df-uni 4577 df-int 4616 df-br 4793 df-opab 4853 df-mpt 4870 df-id 5162 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-iota 6000 df-fun 6039 df-fv 6045 df-1st 7321

This theorem is referenced by: 1stdm 7370

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