Theorem reldmlmd2 50128

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 17829	. 2 ⊢ Rel (𝐷 Func 𝐶)
2		ovex 7400	. . . 4 ⊢ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓) ∈ V
3		lmdfval 50124	. . . 4 ⊢ (𝐶 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓))
4	2, 3	dmmpti 6642	. . 3 ⊢ dom (𝐶 Limit 𝐷) = (𝐷 Func 𝐶)
5	4	releqi 5734	. 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 5631  Rel wrel 5636  ‘cfv 6498  (class class class)co 7367  oppCatcoppc 17677   Func cfunc 17821   FuncCat cfuc 17912  Δ_funccdiag 18178   oppFunc coppf 49597   UP cup 49648   Limit clmd 50118

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-func 17825  df-lmd 50120

This theorem is referenced by: (None)