Theorem nat1st2nd 15189

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 15100	. . . 4 |- Rel ( C Func D )
3		natrcl.1	. . . . . . 7 |- N = ( C Nat D )
4	3	natrcl 15188	. . . . . 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 6827	. . . 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 6827	. . . 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 6295	. 2 |- ( ph -> ( F N G ) = ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
13	1, 12	eleqtrd 2531	1 |- ( ph -> A e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1381   e. wcel 1802   <.cop 4016   Rel wrel 4990   ` cfv 5574  (class class class)co 6277   1stc1st 6779   2ndc2nd 6780   Func cfunc 15092   Nat cnat 15179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-ixp 7468  df-func 15096  df-nat 15181

This theorem is referenced by:  fuccocl  15202  fuclid  15204  fucrid  15205  fucass  15206  fucsect  15210  invfuc  15212  fucpropd  15215  evlfcllem  15359  evlfcl  15360  curfuncf  15376  yonedalem3a  15412  yonedalem3b  15417  yonedainv  15419  yonffthlem  15420