Theorem f1ores 6839

Description: The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)

Assertion

Ref	Expression
f1ores	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))

Proof of Theorem f1ores

Step	Hyp	Ref	Expression
1		f1ssres 6787	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
2		f1f1orn 6836	. . 3 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → (𝐹 ↾ 𝐶):𝐶–1-1-onto→ran (𝐹 ↾ 𝐶))
3	1, 2	syl 18	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→ran (𝐹 ↾ 𝐶))
4		df-ima 5676	. . 3 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶)
5		f1oeq3 6814	. . 3 ⊢ ((𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶) → ((𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶) ↔ (𝐹 ↾ 𝐶):𝐶–1-1-onto→ran (𝐹 ↾ 𝐶)))
6	4, 5	ax-mp 5	. 2 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶) ↔ (𝐹 ↾ 𝐶):𝐶–1-1-onto→ran (𝐹 ↾ 𝐶))
7	3, 6	sylibr 237	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ⊆ wss 3906  ran crn 5664   ↾ cres 5665   “ cima 5666  –1-1→wf1 6537  –1-1-onto→wf1o 6539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547

This theorem is used by:  f1imacnv  6841  f1oresrab  7127  f1ocoima  7310  isores3  7342  isoini2  7346  f1imaeng  9017  f1imaen2g  9018  f1imaen3g  9019  domunsncan  9072  ssfiALT  9165  f1imaenfi  9186  php3  9200  infdifsn  9633  infxpenlem  10013  ackbij2lem2  10238  fin1a2lem6  10404  grothomex  10829  fsumss  15799  ackbijnn  15905  fprodss  16025  unbenlem  16990  eqgen  19293  symgfixelsi  19549  gsumval3lem1  20019  gsumval3lem2  20020  gsumzaddlem  20035  lindsmm  22028  coe1mul2lem2  22479  tsmsf1o  24353  ovoliunlem1  25712  dvcnvrelem2  26228  logf1o2  26866  dvlog  26867  ushgredgedg  29637  ushgredgedgloop  29639  trlreslem  30109  adjbd1o  32508  rinvf1o  33046  padct  33133  hashimaf1  33225  indf1ofs  33256  eulerpartgbij  34827  eulerpartlemgh  34833  ballotlemfrc  34982  reprpmtf1o  35078  erdsze2lem2  35733  poimirlem4  38332  poimirlem9  38337  ismtyres  38517  pwfi2f1o  43881  sge0f1o  47154  3f1oss1  47870  f1oresf1o  48085  uhgrimisgrgric  48754