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Theorem rnss 4479
Description: Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
rnss (AB → ran A ⊆ ran B)

Proof of Theorem rnss
StepHypRef Expression
1 cnvss 4423 . . 3 (ABAB)
2 dmss 4449 . . 3 (AB → dom A ⊆ dom B)
31, 2syl 14 . 2 (AB → dom A ⊆ dom B)
4 df-rn 4271 . 2 ran A = dom A
5 df-rn 4271 . 2 ran B = dom B
63, 4, 53sstr4g 2954 1 (AB → ran A ⊆ ran B)
Colors of variables: wff set class
Syntax hints:  wi 4  wss 2885  ccnv 4259  dom cdm 4260  ran crn 4261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728  df-opab 3782  df-cnv 4268  df-dm 4270  df-rn 4271
This theorem is referenced by:  imass1  4615  imass2  4616  ssxpbm  4671  ssxp2  4673  ssrnres  4678  funssxp  4973  fssres  4979  dff2  5224  fliftf  5352  1stcof  5701  2ndcof  5702  smores  5817
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