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Theorem fssres 5009
 Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
Assertion
Ref Expression
fssres ((𝐹:AB 𝐶A) → (𝐹𝐶):𝐶B)

Proof of Theorem fssres
StepHypRef Expression
1 df-f 4849 . . 3 (𝐹:AB ↔ (𝐹 Fn A ran 𝐹B))
2 fnssres 4955 . . . . 5 ((𝐹 Fn A 𝐶A) → (𝐹𝐶) Fn 𝐶)
3 resss 4578 . . . . . . 7 (𝐹𝐶) ⊆ 𝐹
4 rnss 4507 . . . . . . 7 ((𝐹𝐶) ⊆ 𝐹 → ran (𝐹𝐶) ⊆ ran 𝐹)
53, 4ax-mp 7 . . . . . 6 ran (𝐹𝐶) ⊆ ran 𝐹
6 sstr 2947 . . . . . 6 ((ran (𝐹𝐶) ⊆ ran 𝐹 ran 𝐹B) → ran (𝐹𝐶) ⊆ B)
75, 6mpan 400 . . . . 5 (ran 𝐹B → ran (𝐹𝐶) ⊆ B)
82, 7anim12i 321 . . . 4 (((𝐹 Fn A 𝐶A) ran 𝐹B) → ((𝐹𝐶) Fn 𝐶 ran (𝐹𝐶) ⊆ B))
98an32s 502 . . 3 (((𝐹 Fn A ran 𝐹B) 𝐶A) → ((𝐹𝐶) Fn 𝐶 ran (𝐹𝐶) ⊆ B))
101, 9sylanb 268 . 2 ((𝐹:AB 𝐶A) → ((𝐹𝐶) Fn 𝐶 ran (𝐹𝐶) ⊆ B))
11 df-f 4849 . 2 ((𝐹𝐶):𝐶B ↔ ((𝐹𝐶) Fn 𝐶 ran (𝐹𝐶) ⊆ B))
1210, 11sylibr 137 1 ((𝐹:AB 𝐶A) → (𝐹𝐶):𝐶B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ⊆ wss 2911  ran crn 4289   ↾ cres 4290   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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-fun 4847  df-fn 4848  df-f 4849 This theorem is referenced by:  fssres2  5010  fresin  5011  f1ssres  5042  feqresmpt  5170  f2ndf  5789  fseq1p1m1  8726
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