Theorem f1ores 6064

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 6021	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
2		f1f1orn 6061	. . 3 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → (𝐹 ↾ 𝐶):𝐶–1-1-onto→ran (𝐹 ↾ 𝐶))
3	1, 2	syl 17	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→ran (𝐹 ↾ 𝐶))
4		df-ima 5051	. . 3 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶)
5		f1oeq3 6042	. . 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 223	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ⊆ wss 3540  ran crn 5039   ↾ cres 5040   “ cima 5041  –1-1→wf1 5801  –1-1-onto→wf1o 5803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811

This theorem is referenced by:  f1imacnv  6066  f1oresrab  6302  isores3  6485  isoini2  6489  f1imaeng  7902  f1imaen2g  7903  domunsncan  7945  php3  8031  ssfi  8065  infdifsn  8437  infxpenlem  8719  ackbij2lem2  8945  fin1a2lem6  9110  grothomex  9530  fsumss  14303  ackbijnn  14399  fprodss  14517  unbenlem  15450  eqgen  17470  symgfixelsi  17678  gsumval3lem1  18129  gsumval3lem2  18130  gsumzaddlem  18144  coe1mul2lem2  19459  lindsmm  19986  tsmsf1o  21758  ovoliunlem1  23077  dvcnvrelem2  23585  logf1o2  24196  dvlog  24197  eupares  26502  adjbd1o  28328  rinvf1o  28814  padct  28885  indf1ofs  29415  eulerpartgbij  29761  eulerpartlemgh  29767  ballotlemfrc  29915  erdsze2lem2  30440  poimirlem4  32583  poimirlem9  32588  ismtyres  32777  pwfi2f1o  36684  sge0f1o  39275  ushgredgedga  40456  ushgredgedgaloop  40458  trlreslem  40907