Theorem op1st 3091

Description: Extract the first member of an ordered pair.

Hypothesis

Ref	Expression
op1st.1	⊢ A ∈ V

Assertion

Ref	Expression
op1st	⊢ (1st ‘⟨A, B⟩) = A

Proof of Theorem op1st

Step	Hyp	Ref	Expression
1		1stval 3089	. 2 ⊢ (1st ‘⟨A, B⟩) = ∪dom {⟨A, B⟩}
2		op1st.1	. . 3 ⊢ A ∈ V
3	2	op1sta 2635	. 2 ⊢ ∪dom {⟨A, B⟩} = A
4	1, 3	eqtr 1119	1 ⊢ (1st ‘⟨A, B⟩) = A

Syntax hints:   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  ⟨cop 1810  ∪cuni 1919  dom cdm 2410   ‘cfv 2422  1^st c1st 3085

This theorem is referenced by:  1st2val 3097  seqlem1 4662  ruclem16 4900  ruclem18 4902  ruclem20 4904

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-1st 3087

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