Theorem funfvima2d 38469

Description: A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypothesis

Ref	Expression
funfvima2d.1	|- ( ph -> F : A --> B )

Assertion

Ref	Expression
funfvima2d	|- ( ( ph /\ x e. A ) -> ( F ` x ) e. ( F " A ) )

Proof of Theorem funfvima2d

Step	Hyp	Ref	Expression
1		funfvima2d.1	. . . 4 |- ( ph -> F : A --> B )
2		ffun 6048	. . . 4 |- ( F : A --> B -> Fun F )
3	1, 2	syl 17	. . 3 |- ( ph -> Fun F )
4		ssid 3624	. . . . 5 |- A C_ A
5	4	a1i 11	. . . 4 |- ( ph -> A C_ A )
6		fdm 6051	. . . . 5 |- ( F : A --> B -> dom F = A )
7	1, 6	syl 17	. . . 4 |- ( ph -> dom F = A )
8	5, 7	sseqtr4d 3642	. . 3 |- ( ph -> A C_ dom F )
9		funfvima2 6493	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( x e. A -> ( F ` x ) e. ( F " A ) ) )
10	3, 8, 9	syl2anc 693	. 2 |- ( ph -> ( x e. A -> ( F ` x ) e. ( F " A ) ) )
11	10	imp 445	1 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. ( F " A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   dom cdm 5114   "cima 5117   Fun wfun 5882   -->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:  imo72b2lem1  38471