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Theorem fssres 5293
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 5122 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 fnssres 5231 . . . . 5 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹𝐶) Fn 𝐶)
3 resss 4838 . . . . . . 7 (𝐹𝐶) ⊆ 𝐹
4 rnss 4764 . . . . . . 7 ((𝐹𝐶) ⊆ 𝐹 → ran (𝐹𝐶) ⊆ ran 𝐹)
53, 4ax-mp 5 . . . . . 6 ran (𝐹𝐶) ⊆ ran 𝐹
6 sstr 3100 . . . . . 6 ((ran (𝐹𝐶) ⊆ ran 𝐹 ∧ ran 𝐹𝐵) → ran (𝐹𝐶) ⊆ 𝐵)
75, 6mpan 420 . . . . 5 (ran 𝐹𝐵 → ran (𝐹𝐶) ⊆ 𝐵)
82, 7anim12i 336 . . . 4 (((𝐹 Fn 𝐴𝐶𝐴) ∧ ran 𝐹𝐵) → ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
98an32s 557 . . 3 (((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ∧ 𝐶𝐴) → ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
101, 9sylanb 282 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
11 df-f 5122 . 2 ((𝐹𝐶):𝐶𝐵 ↔ ((𝐹𝐶) Fn 𝐶 ∧ ran (𝐹𝐶) ⊆ 𝐵))
1210, 11sylibr 133 1 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶):𝐶𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wss 3066  ran crn 4535  cres 4536   Fn wfn 5113  wf 5114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-fun 5120  df-fn 5121  df-f 5122
This theorem is referenced by:  fssresd  5294  fssres2  5295  fresin  5296  f1ssres  5332  feqresmpt  5468  f2ndf  6116  elmapssres  6560  pmresg  6563  finomni  7005  fseq1p1m1  9867  hmeores  12473  limcdifap  12789  isomninnlem  13214
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