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Theorem brfullfun 33307
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 2828 . 2 ((FullFun𝐹𝐴) = 𝐵𝐵 = (FullFun𝐹𝐴))
2 fullfunfnv 33305 . . 3 FullFun𝐹 Fn V
3 brfullfun.1 . . 3 𝐴 ∈ V
4 fnbrfvb 6712 . . 3 ((FullFun𝐹 Fn V ∧ 𝐴 ∈ V) → ((FullFun𝐹𝐴) = 𝐵𝐴FullFun𝐹𝐵))
52, 3, 4mp2an 688 . 2 ((FullFun𝐹𝐴) = 𝐵𝐴FullFun𝐹𝐵)
6 fullfunfv 33306 . . 3 (FullFun𝐹𝐴) = (𝐹𝐴)
76eqeq2i 2834 . 2 (𝐵 = (FullFun𝐹𝐴) ↔ 𝐵 = (𝐹𝐴))
81, 5, 73bitr3i 302 1 (𝐴FullFun𝐹𝐵𝐵 = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wcel 2105  Vcvv 3495   class class class wbr 5058   Fn wfn 6344  cfv 6349  FullFuncfullfn 33209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-symdif 4218  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-eprel 5459  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fo 6355  df-fv 6357  df-1st 7680  df-2nd 7681  df-txp 33213  df-singleton 33221  df-singles 33222  df-image 33223  df-funpart 33233  df-fullfun 33234
This theorem is referenced by:  dfrecs2  33309  dfrdg4  33310
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