Theorem funfvopi 2853

Description: The second element in an ordered pair member of a function is the function's value.

Hypothesis

Ref	Expression
funopfv.1	⊢ B ∈ V

Assertion

Ref	Expression
funfvopi	⊢ (Fun F → (⟨A, B⟩ ∈ F → (F ‘A) = B))

Proof of Theorem funfvopi

Step	Hyp	Ref	Expression
1		funopfv.1	. . 3 ⊢ B ∈ V
2	1	funbrfv 2852	. 2 ⊢ (Fun F → (AFB → (F ‘A) = B))
3		df-br 2063	. 2 ⊢ (AFB ↔ ⟨A, B⟩ ∈ F)
4	2, 3	syl5ibr 182	1 ⊢ (Fun F → (⟨A, B⟩ ∈ F → (F ‘A) = B))

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054  Fun wfun 2416   ‘cfv 2422

This theorem is referenced by:  funopfvg 2854  fvelima 2859  fvsn 2879  tfrlem2 2950  tz7.44-1 2966  tz7.44-2 2967  tz7.44-3 2968  fundmen 3333  aceq3lem 3555

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-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

metamath.org