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Theorem 1stval2 7076
 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 5100 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 3176 . . . . . 6 𝑥 ∈ V
3 vex 3176 . . . . . 6 𝑦 ∈ V
42, 3op1st 7067 . . . . 5 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
52, 3op1stb 4867 . . . . 5 𝑥, 𝑦⟩ = 𝑥
64, 5eqtr4i 2635 . . . 4 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥, 𝑦
7 fveq2 6103 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = (1st ‘⟨𝑥, 𝑦⟩))
8 inteq 4413 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
98inteqd 4415 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
106, 7, 93eqtr4a 2670 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝐴)
1110exlimivv 1847 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝐴)
121, 11sylbi 206 1 (𝐴 ∈ (V × V) → (1st𝐴) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ∩ cint 4410   × cxp 5036  ‘cfv 5804  1st c1st 7057 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059 This theorem is referenced by:  1stdm  7106
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