Theorem coffth 16596

Description: The composition of two fully faithful functors is fully faithful. (Contributed by Mario Carneiro, 28-Jan-2017.)

Hypotheses

Ref	Expression
coffth.f	|- ( ph -> F e. ( ( C Full D ) i^i ( C Faith D ) ) )
coffth.g	|- ( ph -> G e. ( ( D Full E ) i^i ( D Faith E ) ) )

Assertion

Ref	Expression
coffth	|- ( ph -> ( G o.func F ) e. ( ( C Full E ) i^i ( C Faith E ) ) )

Proof of Theorem coffth

Step	Hyp	Ref	Expression
1		inss1 3833	. . . 4 |- ( ( C Full D ) i^i ( C Faith D ) ) C_ ( C Full D )
2		coffth.f	. . . 4 |- ( ph -> F e. ( ( C Full D ) i^i ( C Faith D ) ) )
3	1, 2	sseldi 3601	. . 3 |- ( ph -> F e. ( C Full D ) )
4		inss1 3833	. . . 4 |- ( ( D Full E ) i^i ( D Faith E ) ) C_ ( D Full E )
5		coffth.g	. . . 4 |- ( ph -> G e. ( ( D Full E ) i^i ( D Faith E ) ) )
6	4, 5	sseldi 3601	. . 3 |- ( ph -> G e. ( D Full E ) )
7	3, 6	cofull 16594	. 2 |- ( ph -> ( G o.func F ) e. ( C Full E ) )
8		inss2 3834	. . . 4 |- ( ( C Full D ) i^i ( C Faith D ) ) C_ ( C Faith D )
9	8, 2	sseldi 3601	. . 3 |- ( ph -> F e. ( C Faith D ) )
10		inss2 3834	. . . 4 |- ( ( D Full E ) i^i ( D Faith E ) ) C_ ( D Faith E )
11	10, 5	sseldi 3601	. . 3 |- ( ph -> G e. ( D Faith E ) )
12	9, 11	cofth 16595	. 2 |- ( ph -> ( G o.func F ) e. ( C Faith E ) )
13	7, 12	elind 3798	1 |- ( ph -> ( G o.func F ) e. ( ( C Full E ) i^i ( C Faith E ) ) )

Syntax hints:   -> wi 4   e. wcel 1990   i^i cin 3573  (class class class)co 6650   o._func ccofu 16516   Full cful 16562   Faith cfth 16563

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-rmo 2920  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-cat 16329  df-cid 16330  df-func 16518  df-cofu 16520  df-full 16564  df-fth 16565

This theorem is referenced by: (None)