Theorem funfvop 2857

Description: Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42.

Hypothesis

Ref	Expression
funfvop.1	⊢ B ∈ V

Assertion

Ref	Expression
funfvop	⊢ ((Fun F ∧ A ∈ dom F) → ((F ‘A) = B ↔ ⟨A, B⟩ ∈ F))

Proof of Theorem funfvop

Step	Hyp	Ref	Expression
1		funfvop.1	. . 3 ⊢ B ∈ V
2	1	fnfvop 2856	. 2 ⊢ ((F Fn dom F ∧ A ∈ dom F) → ((F ‘A) = B ↔ ⟨A, B⟩ ∈ F))
3		df-fn 2433	. . 3 ⊢ (F Fn dom F ↔ (Fun F ∧ dom F = dom F))
4		cleqid 1102	. . 3 ⊢ dom F = dom F
5	3, 4	mpbiranr 548	. 2 ⊢ (F Fn dom F ↔ Fun F)
6	2, 5	sylanbr 345	1 ⊢ ((Fun F ∧ A ∈ dom F) → ((F ‘A) = B ↔ ⟨A, B⟩ ∈ F))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810  dom cdm 2410  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  dmfco 2864  fvco 2865  funopfv 2886

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-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

metamath.org