Theorem reldmxpc 48925

Description: The binary product of categories is a proper operator, so it can be used with ovprc1 7468, elbasov 17250, strov2rcl 17251, and so on. See reldmxpcALT 48926 for an alternate proof with less "essential steps" but more "bytes". (Proposed by SN, 15-Oct-2025.) (Contributed by Zhi Wang, 15-Oct-2025.)

Assertion

Ref	Expression
reldmxpc	⊢ Rel dom ×c

Proof of Theorem reldmxpc

Step	Hyp	Ref	Expression
1		relxp 5701	. 2 ⊢ Rel (V × V)
2		fnxpc 18217	. . . 4 ⊢ ×c Fn (V × V)
3	2	fndmi 6670	. . 3 ⊢ dom ×c = (V × V)
4	3	releqi 5785	. 2 ⊢ (Rel dom ×c ↔ Rel (V × V))
5	1, 4	mpbir 231	1 ⊢ Rel dom ×c

Syntax hints:  Vcvv 3479   × cxp 5681  dom cdm 5683  Rel wrel 5688   ×_c cxpc 18209

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430  ax-un 7751

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-fv 6567  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-xpc 18213

This theorem is referenced by:  elxpcbasex1  48927  elxpcbasex2  48929