Theorem elrnmptd 39366

Description: The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
elrnmptd.f	|- F = ( x e. A |-> B )
elrnmptd.x	|- ( ph -> E. x e. A C = B )
elrnmptd.c	|- ( ph -> C e. V )

Assertion

Ref	Expression
elrnmptd	|- ( ph -> C e. ran F )

Distinct variable group:   x, C

Allowed substitution hints:   ph( x)   A( x)   B( x)   F( x)   V( x)

Proof of Theorem elrnmptd

Step	Hyp	Ref	Expression
1		elrnmptd.x	. 2 |- ( ph -> E. x e. A C = B )
2		elrnmptd.c	. . 3 |- ( ph -> C e. V )
3		elrnmptd.f	. . . 4 |- F = ( x e. A |-> B )
4	3	elrnmpt 5372	. . 3 |- ( C e. V -> ( C e. ran F <-> E. x e. A C = B ) )
5	2, 4	syl 17	. 2 |- ( ph -> ( C e. ran F <-> E. x e. A C = B ) )
6	1, 5	mpbird 247	1 |- ( ph -> C e. ran F )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913   |-> 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:  infnsuprnmpt  39465  supminfrnmpt  39672  supminfxrrnmpt  39701  sge0sup  40608  sge0resplit  40623  sge0xaddlem2  40651  sge0pnfmpt  40662  sge0reuz  40664  sge0reuzb  40665  hoidmvlelem2  40810