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Theorem brfullfun 32180
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 2658 . 2 ((FullFun𝐹𝐴) = 𝐵𝐵 = (FullFun𝐹𝐴))
2 fullfunfnv 32178 . . 3 FullFun𝐹 Fn V
3 brfullfun.1 . . 3 𝐴 ∈ V
4 fnbrfvb 6274 . . 3 ((FullFun𝐹 Fn V ∧ 𝐴 ∈ V) → ((FullFun𝐹𝐴) = 𝐵𝐴FullFun𝐹𝐵))
52, 3, 4mp2an 708 . 2 ((FullFun𝐹𝐴) = 𝐵𝐴FullFun𝐹𝐵)
6 fullfunfv 32179 . . 3 (FullFun𝐹𝐴) = (𝐹𝐴)
76eqeq2i 2663 . 2 (𝐵 = (FullFun𝐹𝐴) ↔ 𝐵 = (𝐹𝐴))
81, 5, 73bitr3i 290 1 (𝐴FullFun𝐹𝐵𝐵 = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wcel 2030  Vcvv 3231   class class class wbr 4685   Fn wfn 5921  cfv 5926  FullFuncfullfn 32082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-symdif 3877  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-eprel 5058  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-1st 7210  df-2nd 7211  df-txp 32086  df-singleton 32094  df-singles 32095  df-image 32096  df-funpart 32106  df-fullfun 32107
This theorem is referenced by:  dfrecs2  32182  dfrdg4  32183
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