Theorem reldmlmd2 50006

Description: The domain of (𝐶 Limit 𝐷) is a relation. (Contributed by Zhi Wang, 14-Nov-2025.)

Assertion

Ref	Expression
reldmlmd2	⊢ Rel dom (𝐶 Limit 𝐷)

Proof of Theorem reldmlmd2

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relfunc 17798	. 2 ⊢ Rel (𝐷 Func 𝐶)
2		ovex 7401	. . . 4 ⊢ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓) ∈ V
3		lmdfval 50002	. . . 4 ⊢ (𝐶 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓))
4	2, 3	dmmpti 6644	. . 3 ⊢ dom (𝐶 Limit 𝐷) = (𝐷 Func 𝐶)
5	4	releqi 5735	. 2 ⊢ (Rel dom (𝐶 Limit 𝐷) ↔ Rel (𝐷 Func 𝐶))
6	1, 5	mpbir 231	1 ⊢ Rel dom (𝐶 Limit 𝐷)

Syntax hints:  dom cdm 5632  Rel wrel 5637  ‘cfv 6500  (class class class)co 7368  oppCatcoppc 17646   Func cfunc 17790   FuncCat cfuc 17881  Δ_funccdiag 18147   oppFunc coppf 49475   UP cup 49526   Limit clmd 49996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-func 17794  df-lmd 49998

This theorem is referenced by: (None)