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Theorem fss 4997
Description: Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fss ((𝐹:AB B𝐶) → 𝐹:A𝐶)

Proof of Theorem fss
StepHypRef Expression
1 sstr2 2946 . . . . 5 (ran 𝐹B → (B𝐶 → ran 𝐹𝐶))
21com12 27 . . . 4 (B𝐶 → (ran 𝐹B → ran 𝐹𝐶))
32anim2d 320 . . 3 (B𝐶 → ((𝐹 Fn A ran 𝐹B) → (𝐹 Fn A ran 𝐹𝐶)))
4 df-f 4849 . . 3 (𝐹:AB ↔ (𝐹 Fn A ran 𝐹B))
5 df-f 4849 . . 3 (𝐹:A𝐶 ↔ (𝐹 Fn A ran 𝐹𝐶))
63, 4, 53imtr4g 194 . 2 (B𝐶 → (𝐹:AB𝐹:A𝐶))
76impcom 116 1 ((𝐹:AB B𝐶) → 𝐹:A𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wss 2911  ran crn 4289   Fn wfn 4840  wf 4841
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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-f 4849
This theorem is referenced by:  fssd  4998  fconst6g  5028  f1ss  5040  ffoss  5101  fsn2  5280  ofco  5671  tposf2  5824  issmo2  5845  smoiso  5858  ssdomg  6194  1fv  8766
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