Theorem 1stval2 6115

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 4660	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2		vex 2724	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2724	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	op1st 6106	. . . . 5 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
5	2, 3	op1stb 4450	. . . . 5 ⊢ ∩ ∩ ⟨𝑥, 𝑦⟩ = 𝑥
6	4, 5	eqtr4i 2188	. . . 4 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ⟨𝑥, 𝑦⟩
7		fveq2 5480	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = (1st ‘⟨𝑥, 𝑦⟩))
8		inteq 3821	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ 𝐴 = ∩ ⟨𝑥, 𝑦⟩)
9	8	inteqd 3823	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐴 = ∩ ∩ ⟨𝑥, 𝑦⟩)
10	6, 7, 9	3eqtr4a 2223	. . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = ∩ ∩ 𝐴)
11	10	exlimivv 1883	. 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 1342  ∃wex 1479   ∈ wcel 2135  Vcvv 2721  ⟨cop 3573  ∩ cint 3818   × cxp 4596  ‘cfv 5182  1^st c1st 6098

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-iota 5147  df-fun 5184  df-fv 5190  df-1st 6100

This theorem is referenced by:  1stdm  6142