Theorem gruf 9633

Description: A Grothendieck universe contains all functions on its elements. (Contributed by Mario Carneiro, 10-Jun-2013.)

Assertion

Ref	Expression
gruf	|- ( ( U e. Univ /\ A e. U /\ F : A --> U ) -> F e. U )

Proof of Theorem gruf

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1063	. . . 4 |- ( ( U e. Univ /\ A e. U /\ F : A --> U ) -> F : A --> U )
2	1	feqmptd 6249	. . 3 |- ( ( U e. Univ /\ A e. U /\ F : A --> U ) -> F = ( x e. A |-> ( F ` x ) ) )
3		fvex 6201	. . . 4 |- ( F ` x ) e. _V
4	3	fnasrn 6411	. . 3 |- ( x e. A |-> ( F ` x ) ) = ran ( x e. A |-> <. x , ( F ` x ) >. )
5	2, 4	syl6eq 2672	. 2 |- ( ( U e. Univ /\ A e. U /\ F : A --> U ) -> F = ran ( x e. A |-> <. x , ( F ` x ) >. ) )
6		simpl1 1064	. . . . 5 |- ( ( ( U e. Univ /\ A e. U /\ F : A --> U ) /\ x e. A ) -> U e. Univ )
7		gruel 9625	. . . . . . 7 |- ( ( U e. Univ /\ A e. U /\ x e. A ) -> x e. U )
8	7	3expa 1265	. . . . . 6 |- ( ( ( U e. Univ /\ A e. U ) /\ x e. A ) -> x e. U )
9	8	3adantl3 1219	. . . . 5 |- ( ( ( U e. Univ /\ A e. U /\ F : A --> U ) /\ x e. A ) -> x e. U )
10		ffvelrn 6357	. . . . . 6 |- ( ( F : A --> U /\ x e. A ) -> ( F ` x ) e. U )
11	10	3ad2antl3 1225	. . . . 5 |- ( ( ( U e. Univ /\ A e. U /\ F : A --> U ) /\ x e. A ) -> ( F ` x ) e. U )
12		gruop 9627	. . . . 5 |- ( ( U e. Univ /\ x e. U /\ ( F ` x ) e. U ) -> <. x , ( F ` x ) >. e. U )
13	6, 9, 11, 12	syl3anc 1326	. . . 4 |- ( ( ( U e. Univ /\ A e. U /\ F : A --> U ) /\ x e. A ) -> <. x , ( F ` x ) >. e. U )
14		eqid 2622	. . . 4 |- ( x e. A |-> <. x , ( F ` x ) >. ) = ( x e. A |-> <. x , ( F ` x ) >. )
15	13, 14	fmptd 6385	. . 3 |- ( ( U e. Univ /\ A e. U /\ F : A --> U ) -> ( x e. A |-> <. x , ( F ` x ) >. ) : A --> U )
16		grurn 9623	. . 3 |- ( ( U e. Univ /\ A e. U /\ ( x e. A |-> <. x , ( F ` x ) >. ) : A --> U ) -> ran ( x e. A |-> <. x , ( F ` x ) >. ) e. U )
17	15, 16	syld3an3 1371	. 2 |- ( ( U e. Univ /\ A e. U /\ F : A --> U ) -> ran ( x e. A |-> <. x , ( F ` x ) >. ) e. U )
18	5, 17	eqeltrd 2701	1 |- ( ( U e. Univ /\ A e. U /\ F : A --> U ) -> F e. U )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   <.cop 4183   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888   Univcgru 9612

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-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-tr 4753  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-map 7859  df-gru 9613

This theorem is referenced by: (None)