Theorem arwrcl 16694

Description: The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypothesis

Ref	Expression
arwrcl.a	|- A = (Nat ` C )

Assertion

Ref	Expression
arwrcl	|- ( F e. A -> C e. Cat )

Proof of Theorem arwrcl

Step	Hyp	Ref	Expression
1		df-arw 16677	. . 3 |- Nat = ( c e. Cat |-> U. ran (Homa ` c ) )
2	1	dmmptss 5631	. 2 |- dom Nat C_ Cat
3		elfvdm 6220	. . 3 |- ( F e. (Nat ` C ) -> C e. dom Nat )
4		arwrcl.a	. . 3 |- A = (Nat ` C )
5	3, 4	eleq2s 2719	. 2 |- ( F e. A -> C e. dom Nat )
6	2, 5	sseldi 3601	1 |- ( F e. A -> C e. Cat )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   U.cuni 4436   dom cdm 5114   ran crn 5115   ` cfv 5888   Catccat 16325  Natcarw 16672  Hom_achoma 16673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-arw 16677

This theorem is referenced by:  arwhoma  16695  coafval  16714