Theorem fnbrfvb 6146

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 2610	. . . 4 ⊢ (𝐹‘𝐵) = (𝐹‘𝐵)
2		fvex 6113	. . . . 5 ⊢ (𝐹‘𝐵) ∈ V
3		eqeq2 2621	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → ((𝐹‘𝐵) = 𝑥 ↔ (𝐹‘𝐵) = (𝐹‘𝐵)))
4		breq2 4587	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝐵) → (𝐵𝐹𝑥 ↔ 𝐵𝐹(𝐹‘𝐵)))
5	3, 4	bibi12d 334	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝐵) → (((𝐹‘𝐵) = 𝑥 ↔ 𝐵𝐹𝑥) ↔ ((𝐹‘𝐵) = (𝐹‘𝐵) ↔ 𝐵𝐹(𝐹‘𝐵))))
6	5	imbi2d 329	. . . . 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 5909	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑥 𝐵𝐹𝑥)
8		tz6.12c 6123	. . . . . 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 3232	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = (𝐹‘𝐵) ↔ 𝐵𝐹(𝐹‘𝐵)))
11	1, 10	mpbii 222	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵𝐹(𝐹‘𝐵))
12		breq2 4587	. . 3 ⊢ ((𝐹‘𝐵) = 𝐶 → (𝐵𝐹(𝐹‘𝐵) ↔ 𝐵𝐹𝐶))
13	11, 12	syl5ibcom 234	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 → 𝐵𝐹𝐶))
14		fnfun 5902	. . . 4 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
15		funbrfv 6144	. . . 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 201	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃!weu 2458   class class class wbr 4583  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812

This theorem is referenced by:  fnopfvb  6147  funbrfvb  6148  dffn5  6151  feqmptdf  6161  fnsnfv  6168  fndmdif  6229  dffo4  6283  dff13  6416  isomin  6487  isoini  6488  1stconst  7152  2ndconst  7153  fsplit  7169  seqomlem3  7434  seqomlem4  7435  nqerrel  9633  imasleval  16024  znleval  19722  axcontlem5  25648  elnlfn  28171  adjbd1o  28328  fcoinvbr  28799  br1steq  30917  br2ndeq  30918  fv1stcnv  30925  fv2ndcnv  30926  trpredpred  30972  fvbigcup  31179  fvsingle  31197  imageval  31207  brfullfun  31225  bj-mptval  32251  unccur  32562  poimirlem2  32581  poimirlem23  32602  pw2f1ocnv  36622  brcoffn  37348  funressnfv  39857  fnbrafvb  39883