Theorem rnmptssrn 39368

Description: Inclusion relation for two ranges expressed in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
rnmptssrn.b	|- ( ( ph /\ x e. A ) -> B e. V )
rnmptssrn.y	|- ( ( ph /\ x e. A ) -> E. y e. C B = D )

Assertion

Ref	Expression
rnmptssrn	|- ( ph -> ran ( x e. A |-> B ) C_ ran ( y e. C |-> D ) )

Distinct variable groups:   x, A   y, B   x, C   x, D   ph, x   x, y

Allowed substitution hints:   ph( y)   A( y)   B( x)   C( y)   D( y)   V( x, y)

Proof of Theorem rnmptssrn

Step	Hyp	Ref	Expression
1		rnmptssrn.y	. . . 4 |- ( ( ph /\ x e. A ) -> E. y e. C B = D )
2		rnmptssrn.b	. . . . 5 |- ( ( ph /\ x e. A ) -> B e. V )
3		eqid 2622	. . . . . 6 |- ( y e. C |-> D ) = ( y e. C |-> D )
4	3	elrnmpt 5372	. . . . 5 |- ( B e. V -> ( B e. ran ( y e. C |-> D ) <-> E. y e. C B = D ) )
5	2, 4	syl 17	. . . 4 |- ( ( ph /\ x e. A ) -> ( B e. ran ( y e. C |-> D ) <-> E. y e. C B = D ) )
6	1, 5	mpbird 247	. . 3 |- ( ( ph /\ x e. A ) -> B e. ran ( y e. C |-> D ) )
7	6	ralrimiva 2966	. 2 |- ( ph -> A. x e. A B e. ran ( y e. C |-> D ) )
8		eqid 2622	. . 3 |- ( x e. A |-> B ) = ( x e. A |-> B )
9	8	rnmptss 6392	. 2 |- ( A. x e. A B e. ran ( y e. C |-> D ) -> ran ( x e. A |-> B ) C_ ran ( y e. C |-> D ) )
10	7, 9	syl 17	1 |- ( ph -> ran ( x e. A |-> B ) C_ ran ( y e. C |-> D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   |-> 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-ne 2795  df-ral 2917  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-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  sge0f1o  40599