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Theorem 1stval2 6021
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 4571 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 2663 . . . . . 6 𝑥 ∈ V
3 vex 2663 . . . . . 6 𝑦 ∈ V
42, 3op1st 6012 . . . . 5 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
52, 3op1stb 4369 . . . . 5 𝑥, 𝑦⟩ = 𝑥
64, 5eqtr4i 2141 . . . 4 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥, 𝑦
7 fveq2 5389 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = (1st ‘⟨𝑥, 𝑦⟩))
8 inteq 3744 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
98inteqd 3746 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → 𝐴 = 𝑥, 𝑦⟩)
106, 7, 93eqtr4a 2176 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝐴)
1110exlimivv 1852 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (1st𝐴) = 𝐴)
121, 11sylbi 120 1 (𝐴 ∈ (V × V) → (1st𝐴) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wex 1453  wcel 1465  Vcvv 2660  cop 3500   cint 3741   × cxp 4507  cfv 5093  1st c1st 6004
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fv 5101  df-1st 6006
This theorem is referenced by:  1stdm  6048
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