Theorem funopfvg 2854

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 1877	. . . . . 6 ⊢ (x = B → ⟨A, x⟩ = ⟨A, B⟩)
2	1	eleq1d 1155	. . . . 5 ⊢ (x = B → (⟨A, x⟩ ∈ F ↔ ⟨A, B⟩ ∈ F))
3		cleq2 1110	. . . . 5 ⊢ (x = B → ((F ‘A) = x ↔ (F ‘A) = B))
4	2, 3	imbi12d 474	. . . 4 ⊢ (x = B → ((⟨A, x⟩ ∈ F → (F ‘A) = x) ↔ (⟨A, B⟩ ∈ F → (F ‘A) = B)))
5	4	imbi2d 464	. . 3 ⊢ (x = B → ((Fun F → (⟨A, x⟩ ∈ F → (F ‘A) = x)) ↔ (Fun F → (⟨A, B⟩ ∈ F → (F ‘A) = B))))
6		visset 1350	. . . 4 ⊢ x ∈ V
7	6	funfvopi 2853	. . 3 ⊢ (Fun F → (⟨A, x⟩ ∈ F → (F ‘A) = x))
8	5, 7	vtoclg 1383	. 2 ⊢ (B ∈ C → (Fun F → (⟨A, B⟩ ∈ F → (F ‘A) = B)))
9	8	imp 277	1 ⊢ ((B ∈ C ∧ Fun F) → (⟨A, B⟩ ∈ F → (F ‘A) = B))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ⟨cop 1810  Fun wfun 2416   ‘cfv 2422

This theorem is referenced by:  fvopab3ig 2869  oprabvalig 3048

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