Theorem fnbrfvb 3759

Description: Equivalence of function value and binary relation.

Hypothesis

Ref	Expression
fnfvbr.1	⊢ C ∈ V

Assertion

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

Proof of Theorem fnbrfvb

Step	Hyp	Ref	Expression
1		fnfvbr.1	. 2 ⊢ C ∈ V
2		eqeq2 1487	. . . 4 ⊢ (x = C → ((F ‘B) = x ↔ (F ‘B) = C))
3		breq2 2628	. . . 4 ⊢ (x = C → (BFx ↔ BFC))
4	2, 3	bibi12d 631	. . 3 ⊢ (x = C → (((F ‘B) = x ↔ BFx) ↔ ((F ‘B) = C ↔ BFC)))
5	4	imbi2d 614	. 2 ⊢ (x = C → (((F Fn A ⋀ B ∈ A) → ((F ‘B) = x ↔ BFx)) ↔ ((F Fn A ⋀ B ∈ A) → ((F ‘B) = C ↔ BFC))))
6		fneu 3598	. . 3 ⊢ ((F Fn A ⋀ B ∈ A) → ∃!x BFx)
7		breq1 2627	. . . . . . 7 ⊢ (y = B → (yFx ↔ BFx))
8	7	eubidv 1388	. . . . . 6 ⊢ (y = B → (∃!x yFx ↔ ∃!x BFx))
9		fveq2 3730	. . . . . . . 8 ⊢ (y = B → (F ‘y) = (F ‘B))
10	9	eqeq1d 1486	. . . . . . 7 ⊢ (y = B → ((F ‘y) = x ↔ (F ‘B) = x))
11	10, 7	bibi12d 631	. . . . . 6 ⊢ (y = B → (((F ‘y) = x ↔ yFx) ↔ ((F ‘B) = x ↔ BFx)))
12	8, 11	imbi12d 628	. . . . 5 ⊢ (y = B → ((∃!x yFx → ((F ‘y) = x ↔ yFx)) ↔ (∃!x BFx → ((F ‘B) = x ↔ BFx))))
13		visset 1816	. . . . . 6 ⊢ y ∈ V
14	13	tz6.12c 3746	. . . . 5 ⊢ (∃!x yFx → ((F ‘y) = x ↔ yFx))
15	12, 14	vtoclg 1850	. . . 4 ⊢ (B ∈ A → (∃!x BFx → ((F ‘B) = x ↔ BFx)))
16	15	adantl 390	. . 3 ⊢ ((F Fn A ⋀ B ∈ A) → (∃!x BFx → ((F ‘B) = x ↔ BFx)))
17	6, 16	mpd 26	. 2 ⊢ ((F Fn A ⋀ B ∈ A) → ((F ‘B) = x ↔ BFx))
18	1, 5, 17	vtocl 1845	1 ⊢ ((F Fn A ⋀ B ∈ A) → ((F ‘B) = C ↔ BFC))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∃!weu 1382  Vcvv 1814   class class class wbr 2624   Fn wfn 3183   ‘cfv 3188

This theorem is referenced by:  fnopfvb 3760  funbrfvb 3761  fnsnfv 3773  dffo4 3826  f1fv 3880  isomin 3905  isoini 3906  2ndconst 4103  adjbd1o 10013  bra11 10036

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

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