Theorem brfullfun 34250

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 2745	. 2 ⊢ ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐵 = (FullFun𝐹‘𝐴))
2		fullfunfnv 34248	. . 3 ⊢ FullFun𝐹 Fn V
3		brfullfun.1	. . 3 ⊢ 𝐴 ∈ V
4		fnbrfvb 6822	. . 3 ⊢ ((FullFun𝐹 Fn V ∧ 𝐴 ∈ V) → ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐴FullFun𝐹𝐵))
5	2, 3, 4	mp2an 689	. 2 ⊢ ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐴FullFun𝐹𝐵)
6		fullfunfv 34249	. . 3 ⊢ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)
7	6	eqeq2i 2751	. 2 ⊢ (𝐵 = (FullFun𝐹‘𝐴) ↔ 𝐵 = (𝐹‘𝐴))
8	1, 5, 7	3bitr3i 301	1 ⊢ (𝐴FullFun𝐹𝐵 ↔ 𝐵 = (𝐹‘𝐴))

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  Vcvv 3432   class class class wbr 5074   Fn wfn 6428  ‘cfv 6433  FullFuncfullfn 34152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-symdif 4176  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-eprel 5495  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-1st 7831  df-2nd 7832  df-txp 34156  df-singleton 34164  df-singles 34165  df-image 34166  df-funpart 34176  df-fullfun 34177

This theorem is referenced by:  dfrecs2  34252  dfrdg4  34253