Theorem 1stval2 7948

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) → (1st ‘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 5693	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 3435	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3435	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op1st 7939	. . . . 5 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
5	2, 3	op1stb 5411	. . . . 5 ⊢ ∩ ∩ ⟨𝑥, 𝑦⟩ = 𝑥
6	4, 5	eqtr4i 2765	. . . 4 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ⟨𝑥, 𝑦⟩
7		fveq2 6827	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = (1st ‘⟨𝑥, 𝑦⟩))
8		inteq 4880	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ 𝐴 = ∩ ⟨𝑥, 𝑦⟩)
9	8	inteqd 4882	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐴 = ∩ ∩ ⟨𝑥, 𝑦⟩)
10	6, 7, 9	3eqtr4a 2800	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = ∩ ∩ 𝐴)
11	10	exlimivv 1939	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = ∩ ∩ 𝐴)
12	1, 11	sylbi 218	1 ⊢ (𝐴 ∈ (V × V) → (1st ‘𝐴) = ∩ ∩ 𝐴)

Syntax hints:   → wi 4   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3431  ⟨cop 4561  ∩ cint 4877   × cxp 5616  ‘cfv 6485  1^st c1st 7929

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 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-iota 6441  df-fun 6487  df-fv 6493  df-1st 7931

This theorem is referenced by:  1stdm  7982