Theorem f1ores 5649

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 5602	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
2		f1f1orn 5645	. . 3 ⊢ ((𝐹 ↾ 𝐶):𝐶–1-1→𝐵 → (𝐹 ↾ 𝐶):𝐶–1-1-onto→ran (𝐹 ↾ 𝐶))
3	1, 2	syl 14	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→ran (𝐹 ↾ 𝐶))
4		df-ima 4782	. . 3 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶)
5		f1oeq3 5624	. . 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 134	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ⊆ wss 3220  ran crn 4770   ↾ cres 4771   “ cima 4772  –1-1→wf1 5369  –1-1-onto→wf1o 5371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379

This theorem is referenced by:  f1imacnv  5651  f1oresrab  5864  isores3  6011  isoini2  6015  rinvf1o  6025  f1imaeng  7069  f1imaen2g  7070  preimaf1ofi  7258  endjusym  7426  dju1p1e2  7539  fisumss  12137  fprodssdc  12335  ballotfilemfrc  13248  ssnnctlemct  13315  eqgen  14007  ushgredgedg  16381  ushgredgedgloop  16383  trlreslem  16544  domomsubct  16945