Theorem funopfvb 3762

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

Hypothesis

Ref	Expression
funbrfvb.1	⊢ B ∈ V

Assertion

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

Proof of Theorem funopfvb

Step	Hyp	Ref	Expression
1		funbrfvb.1	. . 3 ⊢ B ∈ V
2	1	funbrfvb 3761	. 2 ⊢ ((Fun F ⋀ A ∈ dom F) → ((F ‘A) = B ↔ AFB))
3		df-br 2625	. 2 ⊢ (AFB ↔ ⟨A, B⟩ ∈ F)
4	2, 3	syl6bb 538	1 ⊢ ((Fun F ⋀ A ∈ dom F) → ((F ‘A) = B ↔ ⟨A, B⟩ ∈ F))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  Vcvv 1814  ⟨cop 2415   class class class wbr 2624  dom cdm 3176  Fun wfun 3182   ‘cfv 3188

This theorem is referenced by:  dmfco 3779  fvco 3780  funfvop 3809

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

Copyright terms: Public domain