Theorem rnmptpr 39358

Description: Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
rnmptpr.a	|- ( ph -> A e. V )
rnmptpr.b	|- ( ph -> B e. W )
rnmptpr.f	|- F = ( x e. { A , B } |-> C )
rnmptpr.d	|- ( x = A -> C = D )
rnmptpr.e	|- ( x = B -> C = E )

Assertion

Ref	Expression
rnmptpr	|- ( ph -> ran F = { D , E } )

Distinct variable groups:   x, A   x, B   x, D   x, E

Allowed substitution hints:   ph( x)   C( x)   F( x)   V( x)   W( x)

Proof of Theorem rnmptpr

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . 6 |- y e. _V
2		rnmptpr.f	. . . . . . 7 |- F = ( x e. { A , B } |-> C )
3	2	elrnmpt 5372	. . . . . 6 |- ( y e. _V -> ( y e. ran F <-> E. x e. { A , B } y = C ) )
4	1, 3	ax-mp 5	. . . . 5 |- ( y e. ran F <-> E. x e. { A , B } y = C )
5	4	a1i 11	. . . 4 |- ( ph -> ( y e. ran F <-> E. x e. { A , B } y = C ) )
6		rnmptpr.a	. . . . 5 |- ( ph -> A e. V )
7		rnmptpr.b	. . . . 5 |- ( ph -> B e. W )
8		rnmptpr.d	. . . . . . 7 |- ( x = A -> C = D )
9	8	eqeq2d 2632	. . . . . 6 |- ( x = A -> ( y = C <-> y = D ) )
10		rnmptpr.e	. . . . . . 7 |- ( x = B -> C = E )
11	10	eqeq2d 2632	. . . . . 6 |- ( x = B -> ( y = C <-> y = E ) )
12	9, 11	rexprg 4235	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. x e. { A , B } y = C <-> ( y = D \/ y = E ) ) )
13	6, 7, 12	syl2anc 693	. . . 4 |- ( ph -> ( E. x e. { A , B } y = C <-> ( y = D \/ y = E ) ) )
14	1	elpr 4198	. . . . . 6 |- ( y e. { D , E } <-> ( y = D \/ y = E ) )
15	14	bicomi 214	. . . . 5 |- ( ( y = D \/ y = E ) <-> y e. { D , E } )
16	15	a1i 11	. . . 4 |- ( ph -> ( ( y = D \/ y = E ) <-> y e. { D , E } ) )
17	5, 13, 16	3bitrd 294	. . 3 |- ( ph -> ( y e. ran F <-> y e. { D , E } ) )
18	17	alrimiv 1855	. 2 |- ( ph -> A. y ( y e. ran F <-> y e. { D , E } ) )
19		dfcleq 2616	. 2 |- ( ran F = { D , E } <-> A. y ( y e. ran F <-> y e. { D , E } ) )
20	18, 19	sylibr 224	1 |- ( ph -> ran F = { D , E } )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   A.wal 1481   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   {cpr 4179   |-> 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-sbc 3436  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:  sge0pr  40611