Theorem reldmxpc 49736

Description: The binary product of categories is a proper operator, so it can be used with ovprc1 7395, elbasov 17177, strov2rcl 17178, and so on. See reldmxpcALT 49737 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 5636	. 2 ⊢ Rel (V × V)
2		fnxpc 18133	. . . 4 ⊢ ×c Fn (V × V)
3	2	fndmi 6589	. . 3 ⊢ dom ×c = (V × V)
4	3	releqi 5721	. 2 ⊢ (Rel dom ×c ↔ Rel (V × V))
5	1, 4	mpbir 232	1 ⊢ Rel dom ×c

Syntax hints:  Vcvv 3431   × cxp 5616  dom cdm 5618  Rel wrel 5623   ×_c cxpc 18125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-xpc 18129

This theorem is referenced by:  elxpcbasex1  49738  elxpcbasex2  49740