Theorem brfullfun 35926

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 2736	. 2 ⊢ ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐵 = (FullFun𝐹‘𝐴))
2		fullfunfnv 35924	. . 3 ⊢ FullFun𝐹 Fn V
3		brfullfun.1	. . 3 ⊢ 𝐴 ∈ V
4		fnbrfvb 6873	. . 3 ⊢ ((FullFun𝐹 Fn V ∧ 𝐴 ∈ V) → ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐴FullFun𝐹𝐵))
5	2, 3, 4	mp2an 692	. 2 ⊢ ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐴FullFun𝐹𝐵)
6		fullfunfv 35925	. . 3 ⊢ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)
7	6	eqeq2i 2742	. 2 ⊢ (𝐵 = (FullFun𝐹‘𝐴) ↔ 𝐵 = (𝐹‘𝐴))
8	1, 5, 7	3bitr3i 301	1 ⊢ (𝐴FullFun𝐹𝐵 ↔ 𝐵 = (𝐹‘𝐴))

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  Vcvv 3436   class class class wbr 5092   Fn wfn 6477  ‘cfv 6482  FullFuncfullfn 35828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-symdif 4204  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-eprel 5519  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fo 6488  df-fv 6490  df-1st 7924  df-2nd 7925  df-txp 35832  df-singleton 35840  df-singles 35841  df-image 35842  df-funpart 35852  df-fullfun 35853

This theorem is referenced by:  dfrecs2  35928  dfrdg4  35929