Theorem funimassd 39431

Description: Sufficient condition for the image of a function being a subclass. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
funimassd.1	|- F/ x ph
funimassd.2	|- ( ph -> Fun F )
funimassd.3	|- ( ( ph /\ x e. A ) -> ( F ` x ) e. B )

Assertion

Ref	Expression
funimassd	|- ( ph -> ( F " A ) C_ B )

Distinct variable groups:   x, A   x, B   x, F

Allowed substitution hint:   ph( x)

Proof of Theorem funimassd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funimassd.2	. . . . 5 |- ( ph -> Fun F )
2	1	adantr 481	. . . 4 |- ( ( ph /\ y e. ( F " A ) ) -> Fun F )
3		simpr 477	. . . 4 |- ( ( ph /\ y e. ( F " A ) ) -> y e. ( F " A ) )
4		fvelima 6248	. . . 4 |- ( ( Fun F /\ y e. ( F " A ) ) -> E. x e. A ( F ` x ) = y )
5	2, 3, 4	syl2anc 693	. . 3 |- ( ( ph /\ y e. ( F " A ) ) -> E. x e. A ( F ` x ) = y )
6		funimassd.1	. . . . 5 |- F/ x ph
7		nfv 1843	. . . . 5 |- F/ x y e. ( F " A )
8	6, 7	nfan 1828	. . . 4 |- F/ x ( ph /\ y e. ( F " A ) )
9		nfv 1843	. . . 4 |- F/ x y e. B
10		id 22	. . . . . . . . 9 |- ( ( F ` x ) = y -> ( F ` x ) = y )
11	10	eqcomd 2628	. . . . . . . 8 |- ( ( F ` x ) = y -> y = ( F ` x ) )
12	11	3ad2ant3 1084	. . . . . . 7 |- ( ( ph /\ x e. A /\ ( F ` x ) = y ) -> y = ( F ` x ) )
13		funimassd.3	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( F ` x ) e. B )
14	13	3adant3 1081	. . . . . . 7 |- ( ( ph /\ x e. A /\ ( F ` x ) = y ) -> ( F ` x ) e. B )
15	12, 14	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ x e. A /\ ( F ` x ) = y ) -> y e. B )
16	15	3exp 1264	. . . . 5 |- ( ph -> ( x e. A -> ( ( F ` x ) = y -> y e. B ) ) )
17	16	adantr 481	. . . 4 |- ( ( ph /\ y e. ( F " A ) ) -> ( x e. A -> ( ( F ` x ) = y -> y e. B ) ) )
18	8, 9, 17	rexlimd 3026	. . 3 |- ( ( ph /\ y e. ( F " A ) ) -> ( E. x e. A ( F ` x ) = y -> y e. B ) )
19	5, 18	mpd 15	. 2 |- ( ( ph /\ y e. ( F " A ) ) -> y e. B )
20	19	ssd 39252	1 |- ( ph -> ( F " A ) C_ B )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   E.wrex 2913   C_ wss 3574   "cima 5117   Fun wfun 5882   ` 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-fv 5896

This theorem is referenced by: (None)