Theorem reldmxpc 48969

Description: The binary product of categories is a proper operator, so it can be used with ovprc1 7438, elbasov 17220, strov2rcl 17221, and so on. See reldmxpcALT 48970 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 5669	. 2 ⊢ Rel (V × V)
2		fnxpc 18173	. . . 4 ⊢ ×c Fn (V × V)
3	2	fndmi 6638	. . 3 ⊢ dom ×c = (V × V)
4	3	releqi 5753	. 2 ⊢ (Rel dom ×c ↔ Rel (V × V))
5	1, 4	mpbir 231	1 ⊢ Rel dom ×c

Syntax hints:  Vcvv 3457   × cxp 5649  dom cdm 5651  Rel wrel 5656   ×_c cxpc 18165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pr 5399  ax-un 7723

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-tp 4604  df-op 4606  df-uni 4881  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-id 5545  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-fv 6535  df-oprab 7403  df-mpo 7404  df-1st 7982  df-2nd 7983  df-xpc 18169

This theorem is referenced by:  elxpcbasex1  48971  elxpcbasex2  48973