Theorem fnfvbr 2855

Description: Equivalence of function value and binary relation.

Hypothesis

Ref	Expression
fnfvbr.1	⊢ C ∈ V

Assertion

Ref	Expression
fnfvbr	⊢ ((F Fn A ∧ B ∈ A) → ((F ‘B) = C ↔ BFC))

Proof of Theorem fnfvbr

Step	Hyp	Ref	Expression
1		fnfvbr.1	. 2 ⊢ C ∈ V
2		cleq2 1110	. . . 4 ⊢ (x = C → ((F ‘B) = x ↔ (F ‘B) = C))
3		breq2 2066	. . . 4 ⊢ (x = C → (BFx ↔ BFC))
4	2, 3	bibi12d 477	. . 3 ⊢ (x = C → (((F ‘B) = x ↔ BFx) ↔ ((F ‘B) = C ↔ BFC)))
5	4	imbi2d 464	. 2 ⊢ (x = C → (((F Fn A ∧ B ∈ A) → ((F ‘B) = x ↔ BFx)) ↔ ((F Fn A ∧ B ∈ A) → ((F ‘B) = C ↔ BFC))))
6		fneu 2728	. . 3 ⊢ ((F Fn A ∧ B ∈ A) → ∃!x BFx)
7		breq1 2065	. . . . . . 7 ⊢ (y = B → (yFx ↔ BFx))
8	7	bieudv 1013	. . . . . 6 ⊢ (y = B → (∃!x yFx ↔ ∃!x BFx))
9		fveq2 2832	. . . . . . . 8 ⊢ (y = B → (F ‘y) = (F ‘B))
10	9	cleq1d 1109	. . . . . . 7 ⊢ (y = B → ((F ‘y) = x ↔ (F ‘B) = x))
11	10, 7	bibi12d 477	. . . . . 6 ⊢ (y = B → (((F ‘y) = x ↔ yFx) ↔ ((F ‘B) = x ↔ BFx)))
12	8, 11	imbi12d 474	. . . . 5 ⊢ (y = B → ((∃!x yFx → ((F ‘y) = x ↔ yFx)) ↔ (∃!x BFx → ((F ‘B) = x ↔ BFx))))
13		visset 1350	. . . . . 6 ⊢ y ∈ V
14	13	tz6.12c 2846	. . . . 5 ⊢ (∃!x yFx → ((F ‘y) = x ↔ yFx))
15	12, 14	vtoclg 1383	. . . 4 ⊢ (B ∈ A → (∃!x BFx → ((F ‘B) = x ↔ BFx)))
16	15	adantl 305	. . 3 ⊢ ((F Fn A ∧ B ∈ A) → (∃!x BFx → ((F ‘B) = x ↔ BFx)))
17	6, 16	mpd 46	. 2 ⊢ ((F Fn A ∧ B ∈ A) → ((F ‘B) = x ↔ BFx))
18	1, 5, 17	vtocl 1378	1 ⊢ ((F Fn A ∧ B ∈ A) → ((F ‘B) = C ↔ BFC))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  fnfvop 2856  fvelrn 2883  f1fv 2916  isomin 2937  isoini 2938

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

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