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Definition df-apply 33408
Description: Define the application function. See brapply 33473 for its value. (Contributed by Scott Fenton, 12-Apr-2014.)
Assertion
Ref Expression
df-apply Apply = (( Bigcup Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))

Detailed syntax breakdown of Definition df-apply
StepHypRef Expression
1 capply 33380 . 2 class Apply
2 cbigcup 33369 . . . 4 class Bigcup
32, 2ccom 5536 . . 3 class ( Bigcup Bigcup )
4 cvv 3469 . . . . . 6 class V
54, 4cxp 5530 . . . . 5 class (V × V)
6 cep 5441 . . . . . . . 8 class E
74, 6ctxp 33365 . . . . . . 7 class (V ⊗ E )
8 csingles 33374 . . . . . . . . 9 class Singletons
96, 8cres 5534 . . . . . . . 8 class ( E ↾ Singletons )
109, 4ctxp 33365 . . . . . . 7 class (( E ↾ Singletons ) ⊗ V)
117, 10csymdif 4192 . . . . . 6 class ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))
1211crn 5533 . . . . 5 class ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))
135, 12cdif 3905 . . . 4 class ((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V)))
14 csingle 33373 . . . . . 6 class Singleton
15 cimg 33377 . . . . . 6 class Img
1614, 15ccom 5536 . . . . 5 class (Singleton ∘ Img)
17 cid 5436 . . . . . 6 class I
1817, 14cpprod 33366 . . . . 5 class pprod( I , Singleton)
1916, 18ccom 5536 . . . 4 class ((Singleton ∘ Img) ∘ pprod( I , Singleton))
2013, 19ccom 5536 . . 3 class (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton)))
213, 20ccom 5536 . 2 class (( Bigcup Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
221, 21wceq 1538 1 wff Apply = (( Bigcup Bigcup ) ∘ (((V × V) ∖ ran ((V ⊗ E ) △ (( E ↾ Singletons ) ⊗ V))) ∘ ((Singleton ∘ Img) ∘ pprod( I , Singleton))))
Colors of variables: wff setvar class
This definition is referenced by:  brapply  33473
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