Theorem rnmptssd 39385

Description: The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
rnmptssd.1	|- F/ x ph
rnmptssd.2	|- F = ( x e. A |-> B )
rnmptssd.3	|- ( ( ph /\ x e. A ) -> B e. C )

Assertion

Ref	Expression
rnmptssd	|- ( ph -> ran F C_ C )

Distinct variable groups:   x, A   x, C

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

Proof of Theorem rnmptssd

Step	Hyp	Ref	Expression
1		rnmptssd.1	. . 3 |- F/ x ph
2		rnmptssd.3	. . . 4 |- ( ( ph /\ x e. A ) -> B e. C )
3	2	ex 450	. . 3 |- ( ph -> ( x e. A -> B e. C ) )
4	1, 3	ralrimi 2957	. 2 |- ( ph -> A. x e. A B e. C )
5		rnmptssd.2	. . 3 |- F = ( x e. A |-> B )
6	5	rnmptss 6392	. 2 |- ( A. x e. A B e. C -> ran F C_ C )
7	4, 6	syl 17	1 |- ( ph -> ran F C_ C )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   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:  infnsuprnmpt  39465  suprclrnmpt  39466  suprubrnmpt2  39467  suprubrnmpt  39468  fisupclrnmpt  39622  supxrleubrnmpt  39632  infxrlbrnmpt2  39637  supxrrernmpt  39648  suprleubrnmpt  39649  infrnmptle  39650  infxrunb3rnmpt  39655  supxrre3rnmpt  39656  supminfrnmpt  39672  infxrrnmptcl  39675  infxrgelbrnmpt  39683  infrpgernmpt  39695  supminfxrrnmpt  39701  liminfcl  39995  sge0xaddlem2  40651  sge0reuz  40664  sge0reuzb  40665  hoidmvlelem2  40810  iunhoiioolem  40889  vonioolem1  40894  smflimsuplem4  41029