Theorem prcoffunca

Description: The pre-composition functor is a functor. (Contributed by Zhi Wang, 2-Nov-2025)

	Ref	Expression
Hypotheses	prcoffunc.r	$$⊢R=D\mathrm{FuncCat}E$$
	prcoffunc.e	$$⊢\phi \to E\in \mathrm{Cat}$$
	prcoffunc.s	$$⊢S=C\mathrm{FuncCat}E$$
	prcoffunca.f	$$⊢\phi \to F\in \left(C\mathrm{Func}D\right)$$
Assertion	prcoffunca	Could not format assertion : No typesetting found for |- ( ph -> ( <. D , E >. -o.F F ) e. ( R Func S ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		prcoffunc.r	$$⊢R=D\mathrm{FuncCat}E$$
2		prcoffunc.e	$$⊢\phi \to E\in \mathrm{Cat}$$
3		prcoffunc.s	$$⊢S=C\mathrm{FuncCat}E$$
4		prcoffunca.f	$$⊢\phi \to F\in \left(C\mathrm{Func}D\right)$$
5		eqid	$$⊢C\mathrm{FuncCat}D=C\mathrm{FuncCat}D$$
6		eqidd	Could not format ( ph -> ( <. ( C FuncCat D ) , R >. curryF ( ( <. C , D >. o.F E ) o.func ( ( C FuncCat D ) swapF R ) ) ) = ( <. ( C FuncCat D ) , R >. curryF ( ( <. C , D >. o.F E ) o.func ( ( C FuncCat D ) swapF R ) ) ) ) : No typesetting found for |- ( ph -> ( <. ( C FuncCat D ) , R >. curryF ( ( <. C , D >. o.F E ) o.func ( ( C FuncCat D ) swapF R ) ) ) = ( <. ( C FuncCat D ) , R >. curryF ( ( <. C , D >. o.F E ) o.func ( ( C FuncCat D ) swapF R ) ) ) ) with typecode |-
7		eqidd	Could not format ( ph -> ( ( 1st ` ( <. ( C FuncCat D ) , R >. curryF ( ( <. C , D >. o.F E ) o.func ( ( C FuncCat D ) swapF R ) ) ) ) ` F ) = ( ( 1st ` ( <. ( C FuncCat D ) , R >. curryF ( ( <. C , D >. o.F E ) o.func ( ( C FuncCat D ) swapF R ) ) ) ) ` F ) ) : No typesetting found for |- ( ph -> ( ( 1st ` ( <. ( C FuncCat D ) , R >. curryF ( ( <. C , D >. o.F E ) o.func ( ( C FuncCat D ) swapF R ) ) ) ) ` F ) = ( ( 1st ` ( <. ( C FuncCat D ) , R >. curryF ( ( <. C , D >. o.F E ) o.func ( ( C FuncCat D ) swapF R ) ) ) ) ` F ) ) with typecode |-
8	1 2 5 6 7 4	prcoftposcurfucoa	Could not format ( ph -> ( <. D , E >. -o.F F ) = ( ( 1st ` ( <. ( C FuncCat D ) , R >. curryF ( ( <. C , D >. o.F E ) o.func ( ( C FuncCat D ) swapF R ) ) ) ) ` F ) ) : No typesetting found for |- ( ph -> ( <. D , E >. -o.F F ) = ( ( 1st ` ( <. ( C FuncCat D ) , R >. curryF ( ( <. C , D >. o.F E ) o.func ( ( C FuncCat D ) swapF R ) ) ) ) ` F ) ) with typecode |-
9	5 1 6 4 2 8 3	precofcl	Could not format ( ph -> ( <. D , E >. -o.F F ) e. ( R Func S ) ) : No typesetting found for |- ( ph -> ( <. D , E >. -o.F F ) e. ( R Func S ) ) with typecode |-