Theorem brfullfun 36406

Description: A binary relation form condition for the full function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brfullfun.1	⊢ 𝐴 ∈ V
brfullfun.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
brfullfun	⊢ (𝐴FullFun𝐹𝐵 ↔ 𝐵 = (𝐹‘𝐴))

Proof of Theorem brfullfun

Step	Hyp	Ref	Expression
1		eqcom 2768	. 2 ⊢ ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐵 = (FullFun𝐹‘𝐴))
2		fullfunfnv 36404	. . 3 ⊢ FullFun𝐹 Fn V
3		brfullfun.1	. . 3 ⊢ 𝐴 ∈ V
4		fnbrfvb 6931	. . 3 ⊢ ((FullFun𝐹 Fn V ∧ 𝐴 ∈ V) → ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐴FullFun𝐹𝐵))
5	2, 3, 4	mp2an 704	. 2 ⊢ ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐴FullFun𝐹𝐵)
6		fullfunfv 36405	. . 3 ⊢ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)
7	6	eqeq2i 2774	. 2 ⊢ (𝐵 = (FullFun𝐹‘𝐴) ↔ 𝐵 = (𝐹‘𝐴))
8	1, 5, 7	3bitr3i 304	1 ⊢ (𝐴FullFun𝐹𝐵 ↔ 𝐵 = (𝐹‘𝐴))

Syntax hints:   ↔ wb 209   = wceq 1568   ∈ wcel 2141  Vcvv 3453   class class class wbr 5108   Fn wfn 6531  ‘cfv 6536  FullFuncfullfn 36306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-symdif 4205  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1st 7985  df-2nd 7986  df-txp 36310  df-singleton 36318  df-singles 36319  df-image 36320  df-funpart 36330  df-fullfun 36331

This theorem is referenced by:  dfrecs2  36408  dfrdg4  36409