Theorem eldmressn 41203

Description: Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Assertion

Ref	Expression
eldmressn	|- ( B e. dom ( F |` { A } ) -> B = A )

Proof of Theorem eldmressn

Step	Hyp	Ref	Expression
1		elin 3796	. . 3 |- ( B e. ( { A } i^i dom F ) <-> ( B e. { A } /\ B e. dom F ) )
2		elsni 4194	. . . 4 |- ( B e. { A } -> B = A )
3	2	adantr 481	. . 3 |- ( ( B e. { A } /\ B e. dom F ) -> B = A )
4	1, 3	sylbi 207	. 2 |- ( B e. ( { A } i^i dom F ) -> B = A )
5		dmres 5419	. 2 |- dom ( F |` { A } ) = ( { A } i^i dom F )
6	4, 5	eleq2s 2719	1 |- ( B e. dom ( F |` { A } ) -> B = A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   {csn 4177   dom cdm 5114   |` cres 5116

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-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-br 4654  df-opab 4713  df-xp 5120  df-dm 5124  df-res 5126

This theorem is referenced by:  dfdfat2  41211