Theorem fnbrfvb 6878

Description: Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

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

Proof of Theorem fnbrfvb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2733	. . . 4 ⊢ (𝐹‘𝐵) = (𝐹‘𝐵)
2		fvex 6841	. . . . 5 ⊢ (𝐹‘𝐵) ∈ V
3		eqeq2 2745	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → ((𝐹‘𝐵) = 𝑥 ↔ (𝐹‘𝐵) = (𝐹‘𝐵)))
4		breq2 5097	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → (𝐵𝐹𝑥 ↔ 𝐵𝐹(𝐹‘𝐵)))
5	3, 4	bibi12d 345	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝐵) → (((𝐹‘𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹‘𝐵) = (𝐹‘𝐵) ↔ 𝐵𝐹(𝐹‘𝐵))))
6	5	imbi2d 340	. . . . 5 ⊢ (𝑥 = (𝐹‘𝐵) → (((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝑥 ↔ 𝐵𝐹𝑥)) ↔ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = (𝐹‘𝐵) ↔ 𝐵𝐹(𝐹‘𝐵)))))
7		fneu 6596	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑥 𝐵𝐹𝑥)
8		tz6.12c 6850	. . . . . 6 ⊢ (∃!𝑥 𝐵𝐹𝑥 → ((𝐹‘𝐵) = 𝑥 ↔ 𝐵𝐹𝑥))
9	7, 8	syl 17	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝑥 ↔ 𝐵𝐹𝑥))
10	2, 6, 9	vtocl 3512	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = (𝐹‘𝐵) ↔ 𝐵𝐹(𝐹‘𝐵)))
11	1, 10	mpbii 233	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵𝐹(𝐹‘𝐵))
12		breq2 5097	. . 3 ⊢ ((𝐹‘𝐵) = 𝐶 → (𝐵𝐹(𝐹‘𝐵) ↔ 𝐵𝐹𝐶))
13	11, 12	syl5ibcom 245	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 → 𝐵𝐹𝐶))
14		fnfun 6586	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
15		funbrfv 6876	. . . 4 ⊢ (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹‘𝐵) = 𝐶))
16	14, 15	syl 17	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹‘𝐵) = 𝐶))
17	16	adantr 480	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐵𝐹𝐶 → (𝐹‘𝐵) = 𝐶))
18	13, 17	impbid 212	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃!weu 2565   class class class wbr 5093  Fun wfun 6480   Fn wfn 6481  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494

This theorem is referenced by:  fnopfvb  6879  funbrfvb  6881  fnbrfvb2  6883  dffn5  6886  feqmptdf  6898  fnsnfv  6907  fndmdif  6981  dffo4  7042  dff13  7194  isomin  7277  isoini  7278  br1steqg  7949  br2ndeqg  7950  1stconst  8036  2ndconst  8037  fsplit  8053  seqomlem3  8377  seqomlem4  8378  nqerrel  10830  imasleval  17447  znleval  21493  scutun12  27752  madeval2  27795  axcontlem5  28948  elnlfn  31910  adjbd1o  32067  fcoinvbr  32587  fv1stcnv  35842  fv2ndcnv  35843  fvbigcup  35965  fvsingle  35983  imageval  35993  brfullfun  36013  bj-mptval  37182  unccur  37663  poimirlem2  37682  poimirlem23  37703  pw2f1ocnv  43154  tfsconcat0i  43462  tfsconcatrev  43465  brcoffn  44147  funressnfv  47167  fnbrafvb  47278