Theorem fvmptndm 6308

Description: Value of a function given by the "maps to" notation, outside of its domain. (Contributed by AV, 31-Dec-2020.)

Hypothesis

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

Assertion

Ref	Expression
fvmptndm	|- ( -. X e. A -> ( F ` X ) = (/) )

Distinct variable group:   x, A

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

Proof of Theorem fvmptndm

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvmptndm.1	. . 3 |- F = ( x e. A |-> B )
2		df-mpt 4730	. . 3 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
3	1, 2	eqtri 2644	. 2 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
4	3	fvopab4ndm 6307	1 |- ( -. X e. A -> ( F ` X ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   (/)c0 3915   {copab 4712   |-> cmpt 4729   ` 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-dm 5124  df-iota 5851  df-fv 5896

This theorem is referenced by:  bropfvvvvlem  7256  bropfvvvv  7257  homarcl  16678