Theorem reldmxpc 49491

Description: The binary product of categories is a proper operator, so it can be used with ovprc1 7397, elbasov 17143, strov2rcl 17144, and so on. See reldmxpcALT 49492 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 5642	. 2 ⊢ Rel (V × V)
2		fnxpc 18099	. . . 4 ⊢ ×c Fn (V × V)
3	2	fndmi 6596	. . 3 ⊢ dom ×c = (V × V)
4	3	releqi 5727	. 2 ⊢ (Rel dom ×c ↔ Rel (V × V))
5	1, 4	mpbir 231	1 ⊢ Rel dom ×c

Syntax hints:  Vcvv 3440   × cxp 5622  dom cdm 5624  Rel wrel 5629   ×_c cxpc 18091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-xpc 18095

This theorem is referenced by:  elxpcbasex1  49493  elxpcbasex2  49495