Theorem nat1st2nd 15849

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 15760	. . . 4 |- Rel ( C Func D )
3		natrcl.1	. . . . . . 7 |- N = ( C Nat D )
4	3	natrcl 15848	. . . . . 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 461	. . . 4 |- ( ph -> F e. ( C Func D ) )
7		1st2nd 6836	. . . 4 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
8	2, 6, 7	sylancr 668	. . 3 |- ( ph -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
9	5	simprd 465	. . . 4 |- ( ph -> G e. ( C Func D ) )
10		1st2nd 6836	. . . 4 |- ( ( Rel ( C Func D ) /\ G e. ( C Func D ) ) -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
11	2, 9, 10	sylancr 668	. . 3 |- ( ph -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
12	8, 11	oveq12d 6306	. 2 |- ( ph -> ( F N G ) = ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
13	1, 12	eleqtrd 2530	1 |- ( ph -> A e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1443   e. wcel 1886   <.cop 3973   Rel wrel 4838   ` cfv 5581  (class class class)co 6288   1stc1st 6788   2ndc2nd 6789   Func cfunc 15752   Nat cnat 15839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-ixp 7520  df-func 15756  df-nat 15841

This theorem is referenced by:  fuccocl  15862  fuclid  15864  fucrid  15865  fucass  15866  fucsect  15870  invfuc  15872  fucpropd  15875  evlfcllem  16099  evlfcl  16100  curfuncf  16116  yonedalem3a  16152  yonedalem3b  16157  yonedainv  16159  yonffthlem  16160