Theorem f1ores 6777

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 6726	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
2		f1f1orn 6774	. . 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 5627	. . 3 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶)
5		f1oeq3 6753	. . 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 234	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ⊆ wss 3897  ran crn 5615   ↾ cres 5616   “ cima 5617  –1-1→wf1 6478  –1-1-onto→wf1o 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488

This theorem is referenced by:  f1imacnv  6779  f1oresrab  7060  f1ocoima  7237  isores3  7269  isoini2  7273  f1imaeng  8936  f1imaen2g  8937  f1imaen3g  8938  domunsncan  8990  ssfiALT  9083  f1imaenfi  9104  php3  9118  infdifsn  9547  infxpenlem  9904  ackbij2lem2  10130  fin1a2lem6  10296  grothomex  10720  fsumss  15632  ackbijnn  15735  fprodss  15855  unbenlem  16820  eqgen  19093  symgfixelsi  19347  gsumval3lem1  19817  gsumval3lem2  19818  gsumzaddlem  19833  lindsmm  21765  coe1mul2lem2  22182  tsmsf1o  24060  ovoliunlem1  25430  dvcnvrelem2  25950  logf1o2  26586  dvlog  26587  ushgredgedg  29207  ushgredgedgloop  29209  trlreslem  29676  adjbd1o  32065  rinvf1o  32612  padct  32701  hashimaf1  32793  indf1ofs  32847  eulerpartgbij  34385  eulerpartlemgh  34391  ballotlemfrc  34540  reprpmtf1o  34639  erdsze2lem2  35248  poimirlem4  37672  poimirlem9  37677  ismtyres  37856  pwfi2f1o  43137  sge0f1o  46428  3f1oss1  47114  f1oresf1o  47329  uhgrimisgrgric  47970