Theorem resdmdfsn 5324

Description: Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself. (Contributed by AV, 2-Dec-2018.)

Assertion

Ref	Expression
resdmdfsn

Proof of Theorem resdmdfsn

Step	Hyp	Ref	Expression
1		resindm 5323	. 2
2		indif1 3741	. . . . 5
3		incom 3690	. . . . . . 7
4		inv1 3812	. . . . . . 7
5	3, 4	eqtri 2486	. . . . . 6
6	5	difeq1i 3617	. . . . 5
7	2, 6	eqtri 2486	. . . 4
8	7	a1i 11	. . 3
9	8	reseq2d 5278	. 2
10	1, 9	eqtr3d 2500	1

Syntax hints:  ->wi 4  =wceq 1395  cvv 3109  \cdif 3472  i^icin 3474  {csn 4029  domcdm 5004  |`cres 5006  Relwrel 5009

This theorem is referenced by:  funresdfunsn  6113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-dm 5014  df-res 5016