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Theorem brfullfun 33799
 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 (𝐴FullFun𝐹𝐵𝐵 = (𝐹𝐴))

Proof of Theorem brfullfun
StepHypRef Expression
1 eqcom 2765 . 2 ((FullFun𝐹𝐴) = 𝐵𝐵 = (FullFun𝐹𝐴))
2 fullfunfnv 33797 . . 3 FullFun𝐹 Fn V
3 brfullfun.1 . . 3 𝐴 ∈ V
4 fnbrfvb 6706 . . 3 ((FullFun𝐹 Fn V ∧ 𝐴 ∈ V) → ((FullFun𝐹𝐴) = 𝐵𝐴FullFun𝐹𝐵))
52, 3, 4mp2an 691 . 2 ((FullFun𝐹𝐴) = 𝐵𝐴FullFun𝐹𝐵)
6 fullfunfv 33798 . . 3 (FullFun𝐹𝐴) = (𝐹𝐴)
76eqeq2i 2771 . 2 (𝐵 = (FullFun𝐹𝐴) ↔ 𝐵 = (𝐹𝐴))
81, 5, 73bitr3i 304 1 (𝐴FullFun𝐹𝐵𝐵 = (𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3409   class class class wbr 5032   Fn wfn 6330  ‘cfv 6335  FullFuncfullfn 33701 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-symdif 4147  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-eprel 5435  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fo 6341  df-fv 6343  df-1st 7693  df-2nd 7694  df-txp 33705  df-singleton 33713  df-singles 33714  df-image 33715  df-funpart 33725  df-fullfun 33726 This theorem is referenced by:  dfrecs2  33801  dfrdg4  33802
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