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Theorem fssres 5093
 Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
Assertion
Ref Expression
fssres ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)

Proof of Theorem fssres
StepHypRef Expression
1 df-f 4933 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 fnssres 5039 . . . . 5 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹𝐶) Fn 𝐶)
3 resss 4662 . . . . . . 7 (𝐹𝐶) ⊆ 𝐹
4 rnss 4591 . . . . . . 7 ((𝐹𝐶) ⊆ 𝐹 → ran (𝐹𝐶) ⊆ ran 𝐹)
53, 4ax-mp 7 . . . . . 6 ran (𝐹𝐶) ⊆ ran 𝐹
6 sstr 2980 . . . . . 6 ((ran (𝐹𝐶) ⊆ ran 𝐹 ∧ ran 𝐹𝐵) → ran (𝐹𝐶) ⊆ 𝐵)
75, 6mpan 408 . . . . 5 (ran 𝐹𝐵 → ran (𝐹𝐶) ⊆ 𝐵)
82, 7anim12i 325 . . . 4 (((𝐹 Fn 𝐴𝐶𝐴) ∧ ran 𝐹𝐵) → ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
98an32s 510 . . 3 (((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ 𝐶𝐴) → ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
101, 9sylanb 272 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
11 df-f 4933 . 2 ((𝐹𝐶):𝐶𝐵 ↔ ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
1210, 11sylibr 141 1 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ⊆ wss 2944  ran crn 4373   ↾ cres 4374   Fn wfn 4924  ⟶wf 4925 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-fun 4931  df-fn 4932  df-f 4933 This theorem is referenced by:  fssres2  5094  fresin  5095  f1ssres  5126  feqresmpt  5254  f2ndf  5874  fseq1p1m1  9057
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