Theorem f1ores 6796

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 6745	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
2		f1f1orn 6793	. . 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 5644	. . 3 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶)
5		f1oeq3 6772	. . 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 1540   ⊆ wss 3911  ran crn 5632   ↾ cres 5633   “ cima 5634  –1-1→wf1 6496  –1-1-onto→wf1o 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506

This theorem is referenced by:  f1imacnv  6798  f1oresrab  7081  f1ocoima  7260  isores3  7292  isoini2  7296  f1imaeng  8962  f1imaen2g  8963  f1imaen3g  8964  domunsncan  9018  ssfiALT  9115  f1imaenfi  9136  php3  9150  infdifsn  9586  infxpenlem  9942  ackbij2lem2  10168  fin1a2lem6  10334  grothomex  10758  fsumss  15667  ackbijnn  15770  fprodss  15890  unbenlem  16855  eqgen  19089  symgfixelsi  19341  gsumval3lem1  19811  gsumval3lem2  19812  gsumzaddlem  19827  lindsmm  21713  coe1mul2lem2  22130  tsmsf1o  24008  ovoliunlem1  25379  dvcnvrelem2  25899  logf1o2  26535  dvlog  26536  ushgredgedg  29132  ushgredgedgloop  29134  trlreslem  29601  adjbd1o  31987  rinvf1o  32527  padct  32616  indf1ofs  32762  eulerpartgbij  34336  eulerpartlemgh  34342  ballotlemfrc  34491  reprpmtf1o  34590  erdsze2lem2  35164  poimirlem4  37591  poimirlem9  37596  ismtyres  37775  pwfi2f1o  43058  sge0f1o  46353  3f1oss1  47049  f1oresf1o  47264  uhgrimisgrgric  47904