Theorem funbrfv 2852

Description: The second argument of a binary relation on a function is the function's value.

Hypothesis

Ref	Expression
funbrfv.1	⊢ B ∈ V

Assertion

Ref	Expression
funbrfv	⊢ (Fun F → (AFB → (F ‘A) = B))

Proof of Theorem funbrfv

Step	Hyp	Ref	Expression
1		brrelex 2446	. . . 4 ⊢ ((Rel F ∧ AFB) → A ∈ V)
2		funrel 2681	. . . 4 ⊢ (Fun F → Rel F)
3	1, 2	sylan 343	. . 3 ⊢ ((Fun F ∧ AFB) → A ∈ V)
4		funbrfv.1	. . . 4 ⊢ B ∈ V
5		breq1 2065	. . . . . . 7 ⊢ (x = A → (xFy ↔ AFy))
6	5	anbi2d 468	. . . . . 6 ⊢ (x = A → ((Fun F ∧ xFy) ↔ (Fun F ∧ AFy)))
7		fveq2 2832	. . . . . . 7 ⊢ (x = A → (F ‘x) = (F ‘A))
8	7	cleq1d 1109	. . . . . 6 ⊢ (x = A → ((F ‘x) = y ↔ (F ‘A) = y))
9	6, 8	imbi12d 474	. . . . 5 ⊢ (x = A → (((Fun F ∧ xFy) → (F ‘x) = y) ↔ ((Fun F ∧ AFy) → (F ‘A) = y)))
10		breq2 2066	. . . . . . 7 ⊢ (y = B → (AFy ↔ AFB))
11	10	anbi2d 468	. . . . . 6 ⊢ (y = B → ((Fun F ∧ AFy) ↔ (Fun F ∧ AFB)))
12		cleq2 1110	. . . . . 6 ⊢ (y = B → ((F ‘A) = y ↔ (F ‘A) = B))
13	11, 12	imbi12d 474	. . . . 5 ⊢ (y = B → (((Fun F ∧ AFy) → (F ‘A) = y) ↔ ((Fun F ∧ AFB) → (F ‘A) = B)))
14		visset 1350	. . . . . . . 8 ⊢ x ∈ V
15	14	tz6.12-1 2842	. . . . . . 7 ⊢ ((xFy ∧ ∃!y xFy) → (F ‘x) = y)
16		funeu 2685	. . . . . . 7 ⊢ ((Fun F ∧ xFy) → ∃!y xFy)
17	15, 16	sylan2 346	. . . . . 6 ⊢ ((xFy ∧ (Fun F ∧ xFy)) → (F ‘x) = y)
18	17	anabss7 385	. . . . 5 ⊢ ((Fun F ∧ xFy) → (F ‘x) = y)
19	9, 13, 18	vtocl2g 1386	. . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → ((Fun F ∧ AFB) → (F ‘A) = B))
20	4, 19	mpan2 519	. . 3 ⊢ (A ∈ V → ((Fun F ∧ AFB) → (F ‘A) = B))
21	3, 20	mpcom 49	. 2 ⊢ ((Fun F ∧ AFB) → (F ‘A) = B)
22	21	exp 291	1 ⊢ (Fun F → (AFB → (F ‘A) = B))

Syntax hints:   → wi 2   ∧ wa 196  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054  Rel wrel 2415  Fun wfun 2416   ‘cfv 2422

This theorem is referenced by:  funfvopi 2853  cbvfo 2923

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