Theorem reldmxpc 50172

Description: The binary product of categories is a proper operator, so it can be used with ovprc1 7452, elbasov 17308, strov2rcl 17309, and so on. See reldmxpcALT 50173 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 5673	. 2 ⊢ Rel (V × V)
2		fnxpc 18264	. . . 4 ⊢ ×c Fn (V × V)
3	2	fndmi 6636	. . 3 ⊢ dom ×c = (V × V)
4	3	releqi 5758	. 2 ⊢ (Rel dom ×c ↔ Rel (V × V))
5	1, 4	mpbir 234	1 ⊢ Rel dom ×c

This proof depends on syntax axioms:  Vcvv 3450   × cxp 5653  dom cdm 5655  Rel wrel 5660   ×_c cxpc 18256

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-xpc 18260

This theorem is used by:  elxpcbasex1  50174  elxpcbasex2  50176