Theorem mapdom3 8132

Description: Set exponentiation dominates the mantissa. (Contributed by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
mapdom3	|- ( ( A e. V /\ B e. W /\ B =/= (/) ) -> A ~<_ ( A ^m B ) )

Proof of Theorem mapdom3

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3931	. . 3 |- ( B =/= (/) <-> E. x x e. B )
2		oveq1 6657	. . . . . . . . . 10 |- ( y = A -> ( y ^m { x } ) = ( A ^m { x } ) )
3		id 22	. . . . . . . . . 10 |- ( y = A -> y = A )
4	2, 3	breq12d 4666	. . . . . . . . 9 |- ( y = A -> ( ( y ^m { x } ) ~~ y <-> ( A ^m { x } ) ~~ A ) )
5		vex 3203	. . . . . . . . . 10 |- y e. _V
6		vex 3203	. . . . . . . . . 10 |- x e. _V
7	5, 6	mapsnen 8035	. . . . . . . . 9 |- ( y ^m { x } ) ~~ y
8	4, 7	vtoclg 3266	. . . . . . . 8 |- ( A e. V -> ( A ^m { x } ) ~~ A )
9	8	3ad2ant1 1082	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ x e. B ) -> ( A ^m { x } ) ~~ A )
10	9	ensymd 8007	. . . . . 6 |- ( ( A e. V /\ B e. W /\ x e. B ) -> A ~~ ( A ^m { x } ) )
11		simp2 1062	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ x e. B ) -> B e. W )
12		simp3 1063	. . . . . . . . 9 |- ( ( A e. V /\ B e. W /\ x e. B ) -> x e. B )
13	12	snssd 4340	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ x e. B ) -> { x } C_ B )
14		ssdomg 8001	. . . . . . . 8 |- ( B e. W -> ( { x } C_ B -> { x } ~<_ B ) )
15	11, 13, 14	sylc 65	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ x e. B ) -> { x } ~<_ B )
16	6	snnz 4309	. . . . . . . 8 |- { x } =/= (/)
17		simpl 473	. . . . . . . . 9 |- ( ( { x } = (/) /\ A = (/) ) -> { x } = (/) )
18	17	necon3ai 2819	. . . . . . . 8 |- ( { x } =/= (/) -> -. ( { x } = (/) /\ A = (/) ) )
19	16, 18	ax-mp 5	. . . . . . 7 |- -. ( { x } = (/) /\ A = (/) )
20		mapdom2 8131	. . . . . . 7 |- ( ( { x } ~<_ B /\ -. ( { x } = (/) /\ A = (/) ) ) -> ( A ^m { x } ) ~<_ ( A ^m B ) )
21	15, 19, 20	sylancl 694	. . . . . 6 |- ( ( A e. V /\ B e. W /\ x e. B ) -> ( A ^m { x } ) ~<_ ( A ^m B ) )
22		endomtr 8014	. . . . . 6 |- ( ( A ~~ ( A ^m { x } ) /\ ( A ^m { x } ) ~<_ ( A ^m B ) ) -> A ~<_ ( A ^m B ) )
23	10, 21, 22	syl2anc 693	. . . . 5 |- ( ( A e. V /\ B e. W /\ x e. B ) -> A ~<_ ( A ^m B ) )
24	23	3expia 1267	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( x e. B -> A ~<_ ( A ^m B ) ) )
25	24	exlimdv 1861	. . 3 |- ( ( A e. V /\ B e. W ) -> ( E. x x e. B -> A ~<_ ( A ^m B ) ) )
26	1, 25	syl5bi 232	. 2 |- ( ( A e. V /\ B e. W ) -> ( B =/= (/) -> A ~<_ ( A ^m B ) ) )
27	26	3impia 1261	1 |- ( ( A e. V /\ B e. W /\ B =/= (/) ) -> A ~<_ ( A ^m B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653  (class class class)co 6650   ^m cmap 7857   ~~ cen 7952   ~<_ cdom 7953

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-int 4476  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-er 7742  df-map 7859  df-en 7956  df-dom 7957

This theorem is referenced by:  infmap2  9040