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Theorem 1stval2 8031
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 5760 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 3484 . . . . . 6 𝑥 ∈ V
3 vex 3484 . . . . . 6 𝑦 ∈ V
42, 3op1st 8022 . . . . 5 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
52, 3op1stb 5476 . . . . 5 𝑥, 𝑦⟩ = 𝑥
64, 5eqtr4i 2768 . . . 4 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥, 𝑦
7 fveq2 6906 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = (1st ‘⟨𝑥, 𝑦⟩))
8 inteq 4949 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
98inteqd 4951 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
106, 7, 93eqtr4a 2803 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝐴)
1110exlimivv 1932 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝐴)
121, 11sylbi 217 1 (𝐴 ∈ (V × V) → (1st𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cop 4632   cint 4946   × cxp 5683  cfv 6561  1st c1st 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014
This theorem is referenced by:  1stdm  8065
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