Theorem reldmxpc 50699

Description: The binary product of categories is a proper operator, so it can be used with ovprc1 7490, elbasov 17438, strov2rcl 17439, and so on. See reldmxpcALT 50700 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 5676	. 2 ⊢ Rel (V × V)
2		fnxpc 18400	. . . 4 ⊢ ×c Fn (V × V)
3	2	fndmi 6673	. . 3 ⊢ dom ×c = (V × V)
4	3	releqi 5762	. 2 ⊢ (Rel dom ×c ↔ Rel (V × V))
5	1, 4	mpbir 234	1 ⊢ Rel dom ×c

This proof depends on syntax axioms:  Vcvv 3454   × cxp 5656  dom cdm 5658  Rel wrel 5663   ×_c cxpc 18392

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-xpc 18396

This theorem is used by:  elxpcbasex1  50701  elxpcbasex2  50703