Theorem reldmxpcALT 50025

Description: Alternate proof of reldmxpc 50024. (Contributed by Zhi Wang, 15-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
reldmxpcALT	⊢ Rel dom ×c

Proof of Theorem reldmxpcALT

Dummy variables 𝑓 𝑏 𝑔 ℎ 𝑟 𝑠 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-xpc 18223	. 2 ⊢ ×c = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋((Base‘𝑟) × (Base‘𝑠)) / 𝑏⦌⦋(𝑢 ∈ 𝑏, 𝑣 ∈ 𝑏 ↦ (((1st ‘𝑢)(Hom ‘𝑟)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑠)(2nd ‘𝑣)))) / ℎ⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑥)ℎ𝑦), 𝑓 ∈ (ℎ‘𝑥) ↦ ⟨((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑟)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑠)(2nd ‘𝑦))(2nd ‘𝑓))⟩))⟩})
2	1	reldmmpo 7544	1 ⊢ Rel dom ×c

Syntax hints:  Vcvv 3455  ⦋csb 3853  {ctp 4593  ⟨cop 4595   × cxp 5659  dom cdm 5661  Rel wrel 5666  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  1^st c1st 7980  2^nd c2nd 7981  ndxcnx 17248  Basecbs 17264  Hom chom 17316  compcco 17317   ×_c cxpc 18219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-dm 5671  df-oprab 7414  df-mpo 7415  df-xpc 18223

This theorem is referenced by: (None)