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Theorem 1stval2 7338
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 5322 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 3331 . . . . . 6 𝑥 ∈ V
3 vex 3331 . . . . . 6 𝑦 ∈ V
42, 3op1st 7329 . . . . 5 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
52, 3op1stb 5076 . . . . 5 𝑥, 𝑦⟩ = 𝑥
64, 5eqtr4i 2773 . . . 4 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥, 𝑦
7 fveq2 6340 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = (1st ‘⟨𝑥, 𝑦⟩))
8 inteq 4618 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
98inteqd 4620 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
106, 7, 93eqtr4a 2808 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝐴)
1110exlimivv 1997 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝐴)
121, 11sylbi 207 1 (𝐴 ∈ (V × V) → (1st𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1620  wex 1841  wcel 2127  Vcvv 3328  cop 4315   cint 4615   × cxp 5252  cfv 6037  1st c1st 7319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-int 4616  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fv 6045  df-1st 7321
This theorem is referenced by:  1stdm  7370
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