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Theorem 1stval2 7724
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) → (1st𝐴) = 𝐴)

Proof of Theorem 1stval2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 5591 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 3401 . . . . . 6 𝑥 ∈ V
3 vex 3401 . . . . . 6 𝑦 ∈ V
42, 3op1st 7715 . . . . 5 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
52, 3op1stb 5326 . . . . 5 𝑥, 𝑦⟩ = 𝑥
64, 5eqtr4i 2764 . . . 4 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥, 𝑦
7 fveq2 6668 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = (1st ‘⟨𝑥, 𝑦⟩))
8 inteq 4836 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
98inteqd 4838 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
106, 7, 93eqtr4a 2799 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝐴)
1110exlimivv 1938 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝐴)
121, 11sylbi 220 1 (𝐴 ∈ (V × V) → (1st𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wex 1786  wcel 2113  Vcvv 3397  cop 4519   cint 4833   × cxp 5517  cfv 6333  1st c1st 7705
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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-int 4834  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6291  df-fun 6335  df-fv 6341  df-1st 7707
This theorem is referenced by:  1stdm  7757
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