lanfn - Mathbox for Zhi Wang

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Theorem lanfn 49347

Description: Lan is a function on ((V × V) × V). (Contributed by Zhi Wang, 3-Nov-2025.)

**Assertion**
Ref	Expression
lanfn	⊢ Lan Fn ((V × V) × V)

Proof of Theorem lanfn Dummy variables 𝑐 𝑑 𝑒 𝑓 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lan 49345	. 2 ⊢ Lan = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1^st ‘𝑝) / 𝑐⦌⦋(2^nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘_F 𝑓)((𝑑 FuncCat 𝑒)UP(𝑐 FuncCat 𝑒))𝑥)))
2		ovex 7433	. . . . 5 ⊢ (𝑐 Func 𝑑) ∈ V
3		ovex 7433	. . . . 5 ⊢ (𝑐 Func 𝑒) ∈ V
4	2, 3	mpoex 8073	. . . 4 ⊢ (𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘_F 𝑓)((𝑑 FuncCat 𝑒)UP(𝑐 FuncCat 𝑒))𝑥)) ∈ V
5	4	csbex 5279	. . 3 ⊢ ⦋(2^nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘_F 𝑓)((𝑑 FuncCat 𝑒)UP(𝑐 FuncCat 𝑒))𝑥)) ∈ V
6	5	csbex 5279	. 2 ⊢ ⦋(1^st ‘𝑝) / 𝑐⦌⦋(2^nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘_F 𝑓)((𝑑 FuncCat 𝑒)UP(𝑐 FuncCat 𝑒))𝑥)) ∈ V
7	1, 6	fnmpoi 8064	1 ⊢ Lan Fn ((V × V) × V)

Colors of variables: wff setvar class

Syntax hints: Vcvv 3457 ⦋csb 3872 ⟨cop 4605 × cxp 5650 Fn wfn 6523 ‘cfv 6528 (class class class)co 7400 ∈ cmpo 7402 1^st c1st 7981 2^nd c2nd 7982 Func cfunc 17854 FuncCat cfuc 17945 UPcup 48974 −∘_F cprcof 49147 Lanclan 49343

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5247 ax-sep 5264 ax-nul 5274 ax-pow 5333 ax-pr 5400 ax-un 7724

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4882 df-iun 4967 df-br 5118 df-opab 5180 df-mpt 5200 df-id 5546 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-iota 6481 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-ov 7403 df-oprab 7404 df-mpo 7405 df-1st 7983 df-2nd 7984 df-lan 49345

This theorem is referenced by: reldmlan2 49353 lanrcl 49357

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