Theorem nat1st2nd 16611

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 16522	. . . 4 |- Rel ( C Func D )
3		natrcl.1	. . . . . . 7 |- N = ( C Nat D )
4	3	natrcl 16610	. . . . . 6 |- ( A e. ( F N G ) -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) )
5	1, 4	syl 17	. . . . 5 |- ( ph -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) )
6	5	simpld 475	. . . 4 |- ( ph -> F e. ( C Func D ) )
7		1st2nd 7214	. . . 4 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
8	2, 6, 7	sylancr 695	. . 3 |- ( ph -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
9	5	simprd 479	. . . 4 |- ( ph -> G e. ( C Func D ) )
10		1st2nd 7214	. . . 4 |- ( ( Rel ( C Func D ) /\ G e. ( C Func D ) ) -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
11	2, 9, 10	sylancr 695	. . 3 |- ( ph -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
12	8, 11	oveq12d 6668	. 2 |- ( ph -> ( F N G ) = ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
13	1, 12	eleqtrd 2703	1 |- ( ph -> A e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Func cfunc 16514   Nat cnat 16601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-ixp 7909  df-func 16518  df-nat 16603

This theorem is referenced by:  fuccocl  16624  fuclid  16626  fucrid  16627  fucass  16628  fucsect  16632  invfuc  16634  fucpropd  16637  evlfcllem  16861  evlfcl  16862  curfuncf  16878  yonedalem3a  16914  yonedalem3b  16919  yonedainv  16921  yonffthlem  16922