Theorem fvelima2 39475

Description: Function value in an image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Assertion

Ref	Expression
fvelima2	|- ( ( F Fn A /\ B e. ( F " C ) ) -> E. x e. ( A i^i C ) ( F ` x ) = B )

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

Proof of Theorem fvelima2

Step	Hyp	Ref	Expression
1		id 22	. . . 4 |- ( B e. ( F " C ) -> B e. ( F " C ) )
2		elimag 5470	. . . 4 |- ( B e. ( F " C ) -> ( B e. ( F " C ) <-> E. x e. C x F B ) )
3	1, 2	mpbid 222	. . 3 |- ( B e. ( F " C ) -> E. x e. C x F B )
4		df-rex 2918	. . 3 |- ( E. x e. C x F B <-> E. x ( x e. C /\ x F B ) )
5	3, 4	sylib 208	. 2 |- ( B e. ( F " C ) -> E. x ( x e. C /\ x F B ) )
6		fnbr 5993	. . . . . . . . 9 |- ( ( F Fn A /\ x F B ) -> x e. A )
7	6	adantrl 752	. . . . . . . 8 |- ( ( F Fn A /\ ( x e. C /\ x F B ) ) -> x e. A )
8		simprl 794	. . . . . . . 8 |- ( ( F Fn A /\ ( x e. C /\ x F B ) ) -> x e. C )
9	7, 8	elind 3798	. . . . . . 7 |- ( ( F Fn A /\ ( x e. C /\ x F B ) ) -> x e. ( A i^i C ) )
10		fnfun 5988	. . . . . . . . 9 |- ( F Fn A -> Fun F )
11		funbrfv 6234	. . . . . . . . . 10 |- ( Fun F -> ( x F B -> ( F ` x ) = B ) )
12	11	imp 445	. . . . . . . . 9 |- ( ( Fun F /\ x F B ) -> ( F ` x ) = B )
13	10, 12	sylan 488	. . . . . . . 8 |- ( ( F Fn A /\ x F B ) -> ( F ` x ) = B )
14	13	adantrl 752	. . . . . . 7 |- ( ( F Fn A /\ ( x e. C /\ x F B ) ) -> ( F ` x ) = B )
15	9, 14	jca 554	. . . . . 6 |- ( ( F Fn A /\ ( x e. C /\ x F B ) ) -> ( x e. ( A i^i C ) /\ ( F ` x ) = B ) )
16	15	ex 450	. . . . 5 |- ( F Fn A -> ( ( x e. C /\ x F B ) -> ( x e. ( A i^i C ) /\ ( F ` x ) = B ) ) )
17	16	eximdv 1846	. . . 4 |- ( F Fn A -> ( E. x ( x e. C /\ x F B ) -> E. x ( x e. ( A i^i C ) /\ ( F ` x ) = B ) ) )
18	17	imp 445	. . 3 |- ( ( F Fn A /\ E. x ( x e. C /\ x F B ) ) -> E. x ( x e. ( A i^i C ) /\ ( F ` x ) = B ) )
19		df-rex 2918	. . 3 |- ( E. x e. ( A i^i C ) ( F ` x ) = B <-> E. x ( x e. ( A i^i C ) /\ ( F ` x ) = B ) )
20	18, 19	sylibr 224	. 2 |- ( ( F Fn A /\ E. x ( x e. C /\ x F B ) ) -> E. x e. ( A i^i C ) ( F ` x ) = B )
21	5, 20	sylan2 491	1 |- ( ( F Fn A /\ B e. ( F " C ) ) -> E. x e. ( A i^i C ) ( F ` x ) = B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   i^i cin 3573   class class class wbr 4653   "cima 5117   Fun wfun 5882   Fn wfn 5883   ` 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-fv 5896

This theorem is referenced by:  limsupresxr  39998  liminfresxr  39999  liminfvalxr  40015