Theorem reldmxpc 50024

Description: The binary product of categories is a proper operator, so it can be used with ovprc1 7449, elbasov 17271, strov2rcl 17272, and so on. See reldmxpcALT 50025 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 5679	. 2 ⊢ Rel (V × V)
2		fnxpc 18227	. . . 4 ⊢ ×c Fn (V × V)
3	2	fndmi 6639	. . 3 ⊢ dom ×c = (V × V)
4	3	releqi 5764	. 2 ⊢ (Rel dom ×c ↔ Rel (V × V))
5	1, 4	mpbir 234	1 ⊢ Rel dom ×c

Syntax hints:  Vcvv 3455   × cxp 5659  dom cdm 5661  Rel wrel 5666   ×_c cxpc 18219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-xpc 18223

This theorem is referenced by:  elxpcbasex1  50026  elxpcbasex2  50028