Theorem funopfvg 3758

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

Assertion

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

Proof of Theorem funopfvg

Step	Hyp	Ref	Expression
1		opeq2 2492	. . . . . 6 ⊢ (x = B → ⟨A, x⟩ = ⟨A, B⟩)
2	1	eleq1d 1543	. . . . 5 ⊢ (x = B → (⟨A, x⟩ ∈ F ↔ ⟨A, B⟩ ∈ F))
3		eqeq2 1487	. . . . 5 ⊢ (x = B → ((F ‘A) = x ↔ (F ‘A) = B))
4	2, 3	imbi12d 628	. . . 4 ⊢ (x = B → ((⟨A, x⟩ ∈ F → (F ‘A) = x) ↔ (⟨A, B⟩ ∈ F → (F ‘A) = B)))
5	4	imbi2d 614	. . 3 ⊢ (x = B → ((Fun F → (⟨A, x⟩ ∈ F → (F ‘A) = x)) ↔ (Fun F → (⟨A, B⟩ ∈ F → (F ‘A) = B))))
6		visset 1816	. . . 4 ⊢ x ∈ V
7	6	funopfv 3757	. . 3 ⊢ (Fun F → (⟨A, x⟩ ∈ F → (F ‘A) = x))
8	5, 7	vtoclg 1850	. 2 ⊢ (B ∈ C → (Fun F → (⟨A, B⟩ ∈ F → (F ‘A) = B)))
9	8	imp 350	1 ⊢ ((B ∈ C ⋀ Fun F) → (⟨A, B⟩ ∈ F → (F ‘A) = B))

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  ⟨cop 2415  Fun wfun 3182   ‘cfv 3188

This theorem is referenced by:  fvopab3ig 3784  oprabvalig 4030  adjeqt 9854  oprabvaligg 10435

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-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204

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