Theorem brrange 34906

Description: Binary relation form of the range function. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brdomain.1	⊢ 𝐴 ∈ V
brdomain.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
brrange	⊢ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)

Proof of Theorem brrange

Step	Hyp	Ref	Expression
1		brdomain.1	. . 3 ⊢ 𝐴 ∈ V
2		brdomain.2	. . 3 ⊢ 𝐵 ∈ V
3	1, 2	brimage 34898	. 2 ⊢ (𝐴Image(2nd ↾ (V × V))𝐵 ↔ 𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
4		df-range 34840	. . 3 ⊢ Range = Image(2nd ↾ (V × V))
5	4	breqi 5155	. 2 ⊢ (𝐴Range𝐵 ↔ 𝐴Image(2nd ↾ (V × V))𝐵)
6		dfrn5 34745	. . 3 ⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)
7	6	eqeq2i 2746	. 2 ⊢ (𝐵 = ran 𝐴 ↔ 𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
8	3, 5, 7	3bitr4i 303	1 ⊢ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)

Syntax hints:   ↔ wb 205   = wceq 1542   ∈ wcel 2107  Vcvv 3475   class class class wbr 5149   × cxp 5675  ran crn 5678   ↾ cres 5679   “ cima 5680  2^nd c2nd 7974  Imagecimage 34812  Rangecrange 34816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-symdif 4243  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-eprel 5581  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7975  df-2nd 7976  df-txp 34826  df-image 34836  df-range 34840

This theorem is referenced by:  brrangeg  34908  brrestrict  34921