Theorem rnmptsn 33182

Description: The range of a function mapping to singletons. (Contributed by ML, 15-Jul-2020.)

Assertion

Ref	Expression
rnmptsn	|- ran ( x e. A |-> { x } ) = { u | E. x e. A u = { x } }

Distinct variable groups:   u, A   x, u

Allowed substitution hint:   A( x)

Proof of Theorem rnmptsn

Step	Hyp	Ref	Expression
1		df-mpt 4730	. . . 4 |- ( x e. A |-> { x } ) = { <. x , u >. | ( x e. A /\ u = { x } ) }
2	1	rneqi 5352	. . 3 |- ran ( x e. A |-> { x } ) = ran { <. x , u >. | ( x e. A /\ u = { x } ) }
3		rnopab 5370	. . 3 |- ran { <. x , u >. | ( x e. A /\ u = { x } ) } = { u | E. x ( x e. A /\ u = { x } ) }
4	2, 3	eqtri 2644	. 2 |- ran ( x e. A |-> { x } ) = { u | E. x ( x e. A /\ u = { x } ) }
5		df-rex 2918	. . 3 |- ( E. x e. A u = { x } <-> E. x ( x e. A /\ u = { x } ) )
6	5	abbii 2739	. 2 |- { u | E. x e. A u = { x } } = { u | E. x ( x e. A /\ u = { x } ) }
7	4, 6	eqtr4i 2647	1 |- ran ( x e. A |-> { x } ) = { u | E. x e. A u = { x } }

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   {csn 4177   {copab 4712   |-> cmpt 4729   ran crn 5115

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-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  f1omptsnlem  33183  mptsnunlem  33185  dissneqlem  33187