Theorem reldmxpc 49828

Description: The binary product of categories is a proper operator, so it can be used with ovprc1 7430, elbasov 17243, strov2rcl 17244, and so on. See reldmxpcALT 49829 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 5661	. 2 ⊢ Rel (V × V)
2		fnxpc 18199	. . . 4 ⊢ ×c Fn (V × V)
3	2	fndmi 6620	. . 3 ⊢ dom ×c = (V × V)
4	3	releqi 5746	. 2 ⊢ (Rel dom ×c ↔ Rel (V × V))
5	1, 4	mpbir 233	1 ⊢ Rel dom ×c

Syntax hints:  Vcvv 3453   × cxp 5641  dom cdm 5643  Rel wrel 5648   ×_c cxpc 18191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-fv 6524  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-xpc 18195

This theorem is referenced by:  elxpcbasex1  49830  elxpcbasex2  49832