Theorem brfullfun 35453

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 2733	. 2 ⊢ ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐵 = (FullFun𝐹‘𝐴))
2		fullfunfnv 35451	. . 3 ⊢ FullFun𝐹 Fn V
3		brfullfun.1	. . 3 ⊢ 𝐴 ∈ V
4		fnbrfvb 6938	. . 3 ⊢ ((FullFun𝐹 Fn V ∧ 𝐴 ∈ V) → ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐴FullFun𝐹𝐵))
5	2, 3, 4	mp2an 689	. 2 ⊢ ((FullFun𝐹‘𝐴) = 𝐵 ↔ 𝐴FullFun𝐹𝐵)
6		fullfunfv 35452	. . 3 ⊢ (FullFun𝐹‘𝐴) = (𝐹‘𝐴)
7	6	eqeq2i 2739	. 2 ⊢ (𝐵 = (FullFun𝐹‘𝐴) ↔ 𝐵 = (𝐹‘𝐴))
8	1, 5, 7	3bitr3i 301	1 ⊢ (𝐴FullFun𝐹𝐵 ↔ 𝐵 = (𝐹‘𝐴))

Syntax hints:   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Vcvv 3468   class class class wbr 5141   Fn wfn 6532  ‘cfv 6537  FullFuncfullfn 35355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-symdif 4237  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-eprel 5573  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7974  df-2nd 7975  df-txp 35359  df-singleton 35367  df-singles 35368  df-image 35369  df-funpart 35379  df-fullfun 35380

This theorem is referenced by:  dfrecs2  35455  dfrdg4  35456