Theorem 1stval2 6134

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 4673	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 2733	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2733	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op1st 6125	. . . . 5 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
5	2, 3	op1stb 4463	. . . . 5 ⊢ ∩ ∩ ⟨𝑥, 𝑦⟩ = 𝑥
6	4, 5	eqtr4i 2194	. . . 4 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ⟨𝑥, 𝑦⟩
7		fveq2 5496	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = (1st ‘⟨𝑥, 𝑦⟩))
8		inteq 3834	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ 𝐴 = ∩ ⟨𝑥, 𝑦⟩)
9	8	inteqd 3836	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐴 = ∩ ∩ ⟨𝑥, 𝑦⟩)
10	6, 7, 9	3eqtr4a 2229	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = ∩ ∩ 𝐴)
11	10	exlimivv 1889	. 2 ⊢ (∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = ∩ ∩ 𝐴)
12	1, 11	sylbi 120	1 ⊢ (𝐴 ∈ (V × V) → (1st ‘𝐴) = ∩ ∩ 𝐴)

Syntax hints:   → wi 4   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730  ⟨cop 3586  ∩ cint 3831   × cxp 4609  ‘cfv 5198  1^st c1st 6117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-1st 6119

This theorem is referenced by:  1stdm  6161