Theorem elimampt 29438

Description: Membership in the image of a mapping. (Contributed by Thierry Arnoux, 3-Jan-2022.)

Hypotheses

Ref	Expression
elimampt.f	|- F = ( x e. A |-> B )
elimampt.c	|- ( ph -> C e. W )
elimampt.d	|- ( ph -> D C_ A )

Assertion

Ref	Expression
elimampt	|- ( ph -> ( C e. ( F " D ) <-> E. x e. D C = B ) )

Distinct variable groups:   x, A   x, C   x, D

Allowed substitution hints:   ph( x)   B( x)   F( x)   W( x)

Proof of Theorem elimampt

Step	Hyp	Ref	Expression
1		df-ima 5127	. . 3 |- ( F " D ) = ran ( F |` D )
2	1	eleq2i 2693	. 2 |- ( C e. ( F " D ) <-> C e. ran ( F |` D ) )
3		elimampt.d	. . . 4 |- ( ph -> D C_ A )
4		elimampt.f	. . . . . . . 8 |- F = ( x e. A |-> B )
5	4	reseq1i 5392	. . . . . . 7 |- ( F |` D ) = ( ( x e. A |-> B ) |` D )
6		resmpt 5449	. . . . . . 7 |- ( D C_ A -> ( ( x e. A |-> B ) |` D ) = ( x e. D |-> B ) )
7	5, 6	syl5eq 2668	. . . . . 6 |- ( D C_ A -> ( F |` D ) = ( x e. D |-> B ) )
8	7	rneqd 5353	. . . . 5 |- ( D C_ A -> ran ( F |` D ) = ran ( x e. D |-> B ) )
9	8	eleq2d 2687	. . . 4 |- ( D C_ A -> ( C e. ran ( F |` D ) <-> C e. ran ( x e. D |-> B ) ) )
10	3, 9	syl 17	. . 3 |- ( ph -> ( C e. ran ( F |` D ) <-> C e. ran ( x e. D |-> B ) ) )
11		elimampt.c	. . . 4 |- ( ph -> C e. W )
12		eqid 2622	. . . . 5 |- ( x e. D |-> B ) = ( x e. D |-> B )
13	12	elrnmpt 5372	. . . 4 |- ( C e. W -> ( C e. ran ( x e. D |-> B ) <-> E. x e. D C = B ) )
14	11, 13	syl 17	. . 3 |- ( ph -> ( C e. ran ( x e. D |-> B ) <-> E. x e. D C = B ) )
15	10, 14	bitrd 268	. 2 |- ( ph -> ( C e. ran ( F |` D ) <-> E. x e. D C = B ) )
16	2, 15	syl5bb 272	1 |- ( ph -> ( C e. ( F " D ) <-> E. x e. D C = B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   |-> cmpt 4729   ran crn 5115   |` cres 5116   "cima 5117

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-rex 2918  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  reprpmtf1o  30704