Theorem wfximgfd 38463

Description: The value of a function on its domain is in the image of the function. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypotheses

Ref	Expression
wfximgfd.1	|- ( ph -> C e. A )
wfximgfd.2	|- ( ph -> F : A --> B )

Assertion

Ref	Expression
wfximgfd	|- ( ph -> ( F ` C ) e. ( F " A ) )

Proof of Theorem wfximgfd

Step	Hyp	Ref	Expression
1		wfximgfd.2	. . 3 |- ( ph -> F : A --> B )
2		ffn 6045	. . 3 |- ( F : A --> B -> F Fn A )
3	1, 2	syl 17	. 2 |- ( ph -> F Fn A )
4		ssid 3624	. . 3 |- A C_ A
5	4	a1i 11	. 2 |- ( ph -> A C_ A )
6		wfximgfd.1	. 2 |- ( ph -> C e. A )
7		fnfvima 6496	. 2 |- ( ( F Fn A /\ A C_ A /\ C e. A ) -> ( F ` C ) e. ( F " A ) )
8	3, 5, 6, 7	syl3anc 1326	1 |- ( ph -> ( F ` C ) e. ( F " A ) )

Syntax hints:   -> wi 4   e. wcel 1990   C_ wss 3574   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  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-fv 5896

This theorem is referenced by:  imo72b2lem0  38465