Theorem f1ores 5301

Description: The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by set.mm contributors, 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		ffun 5226	. . . . . 6 ⊢ (F:A–→B → Fun F)
2	1	adantr 451	. . . . 5 ⊢ ((F:A–→B ∧ C ⊆ A) → Fun F)
3		fdm 5227	. . . . . . 7 ⊢ (F:A–→B → dom F = A)
4	3	sseq2d 3300	. . . . . 6 ⊢ (F:A–→B → (C ⊆ dom F ↔ C ⊆ A))
5	4	biimpar 471	. . . . 5 ⊢ ((F:A–→B ∧ C ⊆ A) → C ⊆ dom F)
6		fores 5279	. . . . 5 ⊢ ((Fun F ∧ C ⊆ dom F) → (F ↾ C):C–onto→(F “ C))
7	2, 5, 6	syl2anc 642	. . . 4 ⊢ ((F:A–→B ∧ C ⊆ A) → (F ↾ C):C–onto→(F “ C))
8		funres11 5165	. . . 4 ⊢ (Fun ◡F → Fun ◡(F ↾ C))
9	7, 8	anim12i 549	. . 3 ⊢ (((F:A–→B ∧ C ⊆ A) ∧ Fun ◡F) → ((F ↾ C):C–onto→(F “ C) ∧ Fun ◡(F ↾ C)))
10	9	an32s 779	. 2 ⊢ (((F:A–→B ∧ Fun ◡F) ∧ C ⊆ A) → ((F ↾ C):C–onto→(F “ C) ∧ Fun ◡(F ↾ C)))
11		df-f1 4793	. . 3 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F))
12	11	anbi1i 676	. 2 ⊢ ((F:A–1-1→B ∧ C ⊆ A) ↔ ((F:A–→B ∧ Fun ◡F) ∧ C ⊆ A))
13		dff1o3 5293	. 2 ⊢ ((F ↾ C):C–1-1-onto→(F “ C) ↔ ((F ↾ C):C–onto→(F “ C) ∧ Fun ◡(F ↾ C)))
14	10, 12, 13	3imtr4i 257	1 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (F ↾ C):C–1-1-onto→(F “ C))

Syntax hints:   → wi 4   ∧ wa 358   ⊆ wss 3258   “ cima 4723  ^◡ccnv 4772  dom cdm 4773   ↾ cres 4775  Fun wfun 4776  –→wf 4778  –1-1→wf1 4779  –onto→wfo 4780  –1-1-onto→wf1o 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795

This theorem is referenced by:  f1imacnv  5303  isoini2  5499  swapres  5513  xpassen  6058  enpw1pw  6076  ncdisjun  6137  sbthlem1  6204  lenc  6224