Theorem swapffunca

Description: The swap functor is a functor. (Contributed by Zhi Wang, 9-Oct-2025)

	Ref	Expression
Hypotheses	swapfid.c	$$⊢\phi \to C\in \mathrm{Cat}$$
	swapfid.d	$$⊢\phi \to D\in \mathrm{Cat}$$
	swapfid.s	$$⊢S=C{\times }_{c}D$$
	swapfid.t	$$⊢T=D{\times }_{c}C$$
Assertion	swapffunca	Could not format assertion : No typesetting found for |- ( ph -> ( C swapF D ) e. ( S Func T ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		swapfid.c	$$⊢\phi \to C\in \mathrm{Cat}$$
2		swapfid.d	$$⊢\phi \to D\in \mathrm{Cat}$$
3		swapfid.s	$$⊢S=C{\times }_{c}D$$
4		swapfid.t	$$⊢T=D{\times }_{c}C$$
5	1 2	swapfelvv	Could not format ( ph -> ( C swapF D ) e. ( _V X. _V ) ) : No typesetting found for |- ( ph -> ( C swapF D ) e. ( _V X. _V ) ) with typecode |-
6		1st2nd2	Could not format ( ( C swapF D ) e. ( _V X. _V ) -> ( C swapF D ) = <. ( 1st ` ( C swapF D ) ) , ( 2nd ` ( C swapF D ) ) >. ) : No typesetting found for |- ( ( C swapF D ) e. ( _V X. _V ) -> ( C swapF D ) = <. ( 1st ` ( C swapF D ) ) , ( 2nd ` ( C swapF D ) ) >. ) with typecode |-
7	5 6	syl	Could not format ( ph -> ( C swapF D ) = <. ( 1st ` ( C swapF D ) ) , ( 2nd ` ( C swapF D ) ) >. ) : No typesetting found for |- ( ph -> ( C swapF D ) = <. ( 1st ` ( C swapF D ) ) , ( 2nd ` ( C swapF D ) ) >. ) with typecode |-
8	1 2 3 4 7	swapffunc	Could not format ( ph -> ( 1st ` ( C swapF D ) ) ( S Func T ) ( 2nd ` ( C swapF D ) ) ) : No typesetting found for |- ( ph -> ( 1st ` ( C swapF D ) ) ( S Func T ) ( 2nd ` ( C swapF D ) ) ) with typecode |-
9		df-br	Could not format ( ( 1st ` ( C swapF D ) ) ( S Func T ) ( 2nd ` ( C swapF D ) ) <-> <. ( 1st ` ( C swapF D ) ) , ( 2nd ` ( C swapF D ) ) >. e. ( S Func T ) ) : No typesetting found for |- ( ( 1st ` ( C swapF D ) ) ( S Func T ) ( 2nd ` ( C swapF D ) ) <-> <. ( 1st ` ( C swapF D ) ) , ( 2nd ` ( C swapF D ) ) >. e. ( S Func T ) ) with typecode |-
10	8 9	sylib	Could not format ( ph -> <. ( 1st ` ( C swapF D ) ) , ( 2nd ` ( C swapF D ) ) >. e. ( S Func T ) ) : No typesetting found for |- ( ph -> <. ( 1st ` ( C swapF D ) ) , ( 2nd ` ( C swapF D ) ) >. e. ( S Func T ) ) with typecode |-
11	7 10	eqeltrd	Could not format ( ph -> ( C swapF D ) e. ( S Func T ) ) : No typesetting found for |- ( ph -> ( C swapF D ) e. ( S Func T ) ) with typecode |-