Theorem reldmxpc 49904

Description: The binary product of categories is a proper operator, so it can be used with ovprc1 7447, elbasov 17272, strov2rcl 17273, and so on. See reldmxpcALT 49905 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 5677	. 2 ⊢ Rel (V × V)
2		fnxpc 18228	. . . 4 ⊢ ×c Fn (V × V)
3	2	fndmi 6637	. . 3 ⊢ dom ×c = (V × V)
4	3	releqi 5762	. 2 ⊢ (Rel dom ×c ↔ Rel (V × V))
5	1, 4	mpbir 234	1 ⊢ Rel dom ×c

Syntax hints:  Vcvv 3463   × cxp 5657  dom cdm 5659  Rel wrel 5664   ×_c cxpc 18220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-xpc 18224

This theorem is referenced by:  elxpcbasex1  49906  elxpcbasex2  49908