Theorem reldmxpc 49245

Description: The binary product of categories is a proper operator, so it can be used with ovprc1 7379, elbasov 17114, strov2rcl 17115, and so on. See reldmxpcALT 49246 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 5631	. 2 ⊢ Rel (V × V)
2		fnxpc 18069	. . . 4 ⊢ ×c Fn (V × V)
3	2	fndmi 6580	. . 3 ⊢ dom ×c = (V × V)
4	3	releqi 5715	. 2 ⊢ (Rel dom ×c ↔ Rel (V × V))
5	1, 4	mpbir 231	1 ⊢ Rel dom ×c

Syntax hints:  Vcvv 3433   × cxp 5611  dom cdm 5613  Rel wrel 5618   ×_c cxpc 18061

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5231  ax-nul 5241  ax-pr 5367  ax-un 7662

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-iun 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-id 5508  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fv 6484  df-oprab 7344  df-mpo 7345  df-1st 7915  df-2nd 7916  df-xpc 18065

This theorem is referenced by:  elxpcbasex1  49247  elxpcbasex2  49249