Theorem nat1st2nd 14975

Description: Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
natrcl.1	|- N = ( C Nat D )
nat1st2nd.2	|- ( ph -> A e. ( F N G ) )

Assertion

Ref	Expression
nat1st2nd	|- ( ph -> A e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )

Proof of Theorem nat1st2nd

Step	Hyp	Ref	Expression
1		nat1st2nd.2	. 2 |- ( ph -> A e. ( F N G ) )
2		relfunc 14886	. . . 4 |- Rel ( C Func D )
3		natrcl.1	. . . . . . 7 |- N = ( C Nat D )
4	3	natrcl 14974	. . . . . 6 |- ( A e. ( F N G ) -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) )
5	1, 4	syl 16	. . . . 5 |- ( ph -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) )
6	5	simpld 459	. . . 4 |- ( ph -> F e. ( C Func D ) )
7		1st2nd 6725	. . . 4 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
8	2, 6, 7	sylancr 663	. . 3 |- ( ph -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
9	5	simprd 463	. . . 4 |- ( ph -> G e. ( C Func D ) )
10		1st2nd 6725	. . . 4 |- ( ( Rel ( C Func D ) /\ G e. ( C Func D ) ) -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
11	2, 9, 10	sylancr 663	. . 3 |- ( ph -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
12	8, 11	oveq12d 6213	. 2 |- ( ph -> ( F N G ) = ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
13	1, 12	eleqtrd 2542	1 |- ( ph -> A e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   <.cop 3986   Rel wrel 4948   ` cfv 5521  (class class class)co 6195   1stc1st 6680   2ndc2nd 6681   Func cfunc 14878   Nat cnat 14965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-ixp 7369  df-func 14882  df-nat 14967

This theorem is referenced by:  fuccocl  14988  fuclid  14990  fucrid  14991  fucass  14992  fucsect  14996  invfuc  14998  fucpropd  15001  evlfcllem  15145  evlfcl  15146  curfuncf  15162  yonedalem3a  15198  yonedalem3b  15203  yonedainv  15205  yonffthlem  15206