Theorem reldmxpc 50081

Description: The binary product of categories is a proper operator, so it can be used with ovprc1 7458, elbasov 17298, strov2rcl 17299, and so on. See reldmxpcALT 50082 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 5681	. 2 ⊢ Rel (V × V)
2		fnxpc 18254	. . . 4 ⊢ ×c Fn (V × V)
3	2	fndmi 6643	. . 3 ⊢ dom ×c = (V × V)
4	3	releqi 5766	. 2 ⊢ (Rel dom ×c ↔ Rel (V × V))
5	1, 4	mpbir 234	1 ⊢ Rel dom ×c

This proof depends on syntax axioms:  Vcvv 3457   × cxp 5661  dom cdm 5663  Rel wrel 5668   ×_c cxpc 18246

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-xpc 18250

This theorem is used by:  elxpcbasex1  50083  elxpcbasex2  50085