Theorem brelrn 2559

Description: The second argument of a binary relation belongs to its range.

Hypotheses

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

Assertion

Ref	Expression
brelrn	⊢ (ACB → B ∈ ran C)

Proof of Theorem brelrn

Step	Hyp	Ref	Expression
1		brelrn.2	. . . 4 ⊢ B ∈ V
2		brelrn.1	. . . 4 ⊢ A ∈ V
3	1, 2	brcnv 2519	. . 3 ⊢ (B◡CA ↔ ACB)
4	1	breldm 2535	. . 3 ⊢ (B◡CA → B ∈ dom ◡C)
5	3, 4	sylbir 176	. 2 ⊢ (ACB → B ∈ dom ◡C)
6		df-rn 2429	. . 3 ⊢ ran C = dom ◡C
7	6	eleq2i 1153	. 2 ⊢ (B ∈ ran C ↔ B ∈ dom ◡C)
8	5, 7	sylibr 175	1 ⊢ (ACB → B ∈ ran C)

Syntax hints:   → wi 2   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054  ^◡ccnv 2409  dom cdm 2410  ran crn 2411

This theorem is referenced by:  opelrn 2560  cores 2659  funcnv 2703  cbvfo 2923

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426  df-dm 2428  df-rn 2429

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