Theorem brrange 33395

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 33387	. 2 ⊢ (𝐴Image(2nd ↾ (V × V))𝐵 ↔ 𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
4		df-range 33329	. . 3 ⊢ Range = Image(2nd ↾ (V × V))
5	4	breqi 5072	. 2 ⊢ (𝐴Range𝐵 ↔ 𝐴Image(2nd ↾ (V × V))𝐵)
6		dfrn5 33017	. . 3 ⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)
7	6	eqeq2i 2834	. 2 ⊢ (𝐵 = ran 𝐴 ↔ 𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
8	3, 5, 7	3bitr4i 305	1 ⊢ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)

Syntax hints:   ↔ wb 208   = wceq 1537   ∈ wcel 2114  Vcvv 3494   class class class wbr 5066   × cxp 5553  ran crn 5556   ↾ cres 5557   “ cima 5558  2^nd c2nd 7688  Imagecimage 33301  Rangecrange 33305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-symdif 4219  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-eprel 5465  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fo 6361  df-fv 6363  df-1st 7689  df-2nd 7690  df-txp 33315  df-image 33325  df-range 33329

This theorem is referenced by:  brrangeg  33397  brrestrict  33410