Theorem f1ores 6832

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 6781	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
2		f1f1orn 6829	. . 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 5667	. . 3 ⊢ (𝐹 “ 𝐶) = ran (𝐹 ↾ 𝐶)
5		f1oeq3 6808	. . 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 3926  ran crn 5655   ↾ cres 5656   “ cima 5657  –1-1→wf1 6528  –1-1-onto→wf1o 6530

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538

This theorem is referenced by:  f1imacnv  6834  f1oresrab  7117  f1ocoima  7296  isores3  7328  isoini2  7332  f1imaeng  9028  f1imaen2g  9029  f1imaen3g  9030  domunsncan  9086  ssfiALT  9188  f1imaenfi  9209  php3  9223  php3OLD  9233  infdifsn  9671  infxpenlem  10027  ackbij2lem2  10253  fin1a2lem6  10419  grothomex  10843  fsumss  15741  ackbijnn  15844  fprodss  15964  unbenlem  16928  eqgen  19164  symgfixelsi  19416  gsumval3lem1  19886  gsumval3lem2  19887  gsumzaddlem  19902  lindsmm  21788  coe1mul2lem2  22205  tsmsf1o  24083  ovoliunlem1  25455  dvcnvrelem2  25975  logf1o2  26611  dvlog  26612  ushgredgedg  29208  ushgredgedgloop  29210  trlreslem  29679  adjbd1o  32066  rinvf1o  32608  padct  32697  indf1ofs  32843  eulerpartgbij  34404  eulerpartlemgh  34410  ballotlemfrc  34559  reprpmtf1o  34658  erdsze2lem2  35226  poimirlem4  37648  poimirlem9  37653  ismtyres  37832  pwfi2f1o  43120  sge0f1o  46411  3f1oss1  47104  f1oresf1o  47319  uhgrimisgrgric  47944