Theorem mptrcl 6289

Description: Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.)

Hypothesis

Ref	Expression
mptrcl.1	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
mptrcl	|- ( I e. ( F ` X ) -> X e. A )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   F( x)   I( x)   X( x)

Proof of Theorem mptrcl

Step	Hyp	Ref	Expression
1		n0i 3920	. 2 |- ( I e. ( F ` X ) -> -. ( F ` X ) = (/) )
2		mptrcl.1	. . . . 5 |- F = ( x e. A |-> B )
3	2	dmmptss 5631	. . . 4 |- dom F C_ A
4	3	sseli 3599	. . 3 |- ( X e. dom F -> X e. A )
5		ndmfv 6218	. . 3 |- ( -. X e. dom F -> ( F ` X ) = (/) )
6	4, 5	nsyl4 156	. 2 |- ( -. ( F ` X ) = (/) -> X e. A )
7	1, 6	syl 17	1 |- ( I e. ( F ` X ) -> X e. A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   (/)c0 3915   |-> cmpt 4729   dom cdm 5114   ` 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-8 1992  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-pow 4843  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-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-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896

This theorem is referenced by:  initorcl  16644  termorcl  16645  zeroorcl  16646  issubrg  18780  elmptrab  21630