Theorem imadomg 9356

Description: An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92. (Contributed by NM, 23-Jul-2004.)

Assertion

Ref	Expression
imadomg	|- ( A e. B -> ( Fun F -> ( F " A ) ~<_ A ) )

Proof of Theorem imadomg

Step	Hyp	Ref	Expression
1		df-ima 5127	. . . 4 |- ( F " A ) = ran ( F |` A )
2		resfunexg 6479	. . . . . 6 |- ( ( Fun F /\ A e. B ) -> ( F |` A ) e. _V )
3		dmexg 7097	. . . . . 6 |- ( ( F |` A ) e. _V -> dom ( F |` A ) e. _V )
4	2, 3	syl 17	. . . . 5 |- ( ( Fun F /\ A e. B ) -> dom ( F |` A ) e. _V )
5		funres 5929	. . . . . . 7 |- ( Fun F -> Fun ( F |` A ) )
6		funforn 6122	. . . . . . 7 |- ( Fun ( F |` A ) <-> ( F |` A ) : dom ( F |` A ) -onto-> ran ( F |` A ) )
7	5, 6	sylib 208	. . . . . 6 |- ( Fun F -> ( F |` A ) : dom ( F |` A ) -onto-> ran ( F |` A ) )
8	7	adantr 481	. . . . 5 |- ( ( Fun F /\ A e. B ) -> ( F |` A ) : dom ( F |` A ) -onto-> ran ( F |` A ) )
9		fodomg 9345	. . . . 5 |- ( dom ( F |` A ) e. _V -> ( ( F |` A ) : dom ( F |` A ) -onto-> ran ( F |` A ) -> ran ( F |` A ) ~<_ dom ( F |` A ) ) )
10	4, 8, 9	sylc 65	. . . 4 |- ( ( Fun F /\ A e. B ) -> ran ( F |` A ) ~<_ dom ( F |` A ) )
11	1, 10	syl5eqbr 4688	. . 3 |- ( ( Fun F /\ A e. B ) -> ( F " A ) ~<_ dom ( F |` A ) )
12	11	expcom 451	. 2 |- ( A e. B -> ( Fun F -> ( F " A ) ~<_ dom ( F |` A ) ) )
13		dmres 5419	. . . . . 6 |- dom ( F |` A ) = ( A i^i dom F )
14		inss1 3833	. . . . . 6 |- ( A i^i dom F ) C_ A
15	13, 14	eqsstri 3635	. . . . 5 |- dom ( F |` A ) C_ A
16		ssdomg 8001	. . . . 5 |- ( A e. B -> ( dom ( F |` A ) C_ A -> dom ( F |` A ) ~<_ A ) )
17	15, 16	mpi 20	. . . 4 |- ( A e. B -> dom ( F |` A ) ~<_ A )
18		domtr 8009	. . . 4 |- ( ( ( F " A ) ~<_ dom ( F |` A ) /\ dom ( F |` A ) ~<_ A ) -> ( F " A ) ~<_ A )
19	17, 18	sylan2 491	. . 3 |- ( ( ( F " A ) ~<_ dom ( F |` A ) /\ A e. B ) -> ( F " A ) ~<_ A )
20	19	expcom 451	. 2 |- ( A e. B -> ( ( F " A ) ~<_ dom ( F |` A ) -> ( F " A ) ~<_ A ) )
21	12, 20	syld 47	1 |- ( A e. B -> ( Fun F -> ( F " A ) ~<_ A ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -onto->wfo 5886   ~<_ 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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-card 8765  df-acn 8768  df-ac 8939

This theorem is referenced by:  fimact  9357  uniimadom  9366  hausmapdom  21303