Theorem brrange 35929

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 35921	. 2 ⊢ (𝐴Image(2nd ↾ (V × V))𝐵 ↔ 𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
4		df-range 35863	. . 3 ⊢ Range = Image(2nd ↾ (V × V))
5	4	breqi 5116	. 2 ⊢ (𝐴Range𝐵 ↔ 𝐴Image(2nd ↾ (V × V))𝐵)
6		dfrn5 35768	. . 3 ⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)
7	6	eqeq2i 2743	. 2 ⊢ (𝐵 = ran 𝐴 ↔ 𝐵 = ((2nd ↾ (V × V)) “ 𝐴))
8	3, 5, 7	3bitr4i 303	1 ⊢ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  Vcvv 3450   class class class wbr 5110   × cxp 5639  ran crn 5642   ↾ cres 5643   “ cima 5644  2^nd c2nd 7970  Imagecimage 35835  Rangecrange 35839

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-symdif 4219  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-eprel 5541  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fo 6520  df-fv 6522  df-1st 7971  df-2nd 7972  df-txp 35849  df-image 35859  df-range 35863

This theorem is referenced by:  brrangeg  35931  brrestrict  35944