Theorem nat1st2nd 15439

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 15350	. . . 4 |- Rel ( C Func D )
3		natrcl.1	. . . . . . 7 |- N = ( C Nat D )
4	3	natrcl 15438	. . . . . 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 457	. . . 4 |- ( ph -> F e. ( C Func D ) )
7		1st2nd 6819	. . . 4 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
8	2, 6, 7	sylancr 661	. . 3 |- ( ph -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
9	5	simprd 461	. . . 4 |- ( ph -> G e. ( C Func D ) )
10		1st2nd 6819	. . . 4 |- ( ( Rel ( C Func D ) /\ G e. ( C Func D ) ) -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
11	2, 9, 10	sylancr 661	. . 3 |- ( ph -> G = <. ( 1st ` G ) , ( 2nd ` G ) >. )
12	8, 11	oveq12d 6288	. 2 |- ( ph -> ( F N G ) = ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
13	1, 12	eleqtrd 2544	1 |- ( ph -> A e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1398   e. wcel 1823   <.cop 4022   Rel wrel 4993   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   Func cfunc 15342   Nat cnat 15429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-ixp 7463  df-func 15346  df-nat 15431

This theorem is referenced by:  fuccocl  15452  fuclid  15454  fucrid  15455  fucass  15456  fucsect  15460  invfuc  15462  fucpropd  15465  evlfcllem  15689  evlfcl  15690  curfuncf  15706  yonedalem3a  15742  yonedalem3b  15747  yonedainv  15749  yonffthlem  15750