Theorem reldmxpc 49217

Description: The binary product of categories is a proper operator, so it can be used with ovprc1 7428, elbasov 17192, strov2rcl 17193, and so on. See reldmxpcALT 49218 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 5658	. 2 ⊢ Rel (V × V)
2		fnxpc 18143	. . . 4 ⊢ ×c Fn (V × V)
3	2	fndmi 6624	. . 3 ⊢ dom ×c = (V × V)
4	3	releqi 5742	. 2 ⊢ (Rel dom ×c ↔ Rel (V × V))
5	1, 4	mpbir 231	1 ⊢ Rel dom ×c

Syntax hints:  Vcvv 3450   × cxp 5638  dom cdm 5640  Rel wrel 5645   ×_c cxpc 18135

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 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fv 6521  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-xpc 18139

This theorem is referenced by:  elxpcbasex1  49219  elxpcbasex2  49221