Theorem brfullfun 36236

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 2759	. 2 ⊢ ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐵 = (FullFun𝐹‘𝐴))
2		fullfunfnv 36234	. . 3 ⊢ FullFun𝐹 Fn V
3		brfullfun.1	. . 3 ⊢ 𝐴 ∈ V
4		fnbrfvb 6902	. . 3 ⊢ ((FullFun𝐹 Fn V ∧ 𝐴 ∈ V) → ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐴FullFun𝐹𝐵))
5	2, 3, 4	mp2an 700	. 2 ⊢ ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐴FullFun𝐹𝐵)
6		fullfunfv 36235	. . 3 ⊢ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)
7	6	eqeq2i 2765	. 2 ⊢ (𝐵 = (FullFun𝐹‘𝐴) ↔ 𝐵 = (𝐹‘𝐴))
8	1, 5, 7	3bitr3i 303	1 ⊢ (𝐴FullFun𝐹𝐵 ↔ 𝐵 = (𝐹‘𝐴))

Syntax hints:   ↔ wb 208   = wceq 1550   ∈ wcel 2132  Vcvv 3444   class class class wbr 5090   Fn wfn 6501  ‘cfv 6506  FullFuncfullfn 36136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-symdif 4196  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-eprel 5536  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-fo 6512  df-fv 6514  df-1st 7955  df-2nd 7956  df-txp 36140  df-singleton 36148  df-singles 36149  df-image 36150  df-funpart 36160  df-fullfun 36161

This theorem is referenced by:  dfrecs2  36238  dfrdg4  36239