Theorem f1ores 5300

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 5225	. . . . . 6 ⊢ (F:A–→B → Fun F)
2	1	adantr 451	. . . . 5 ⊢ ((F:A–→B ∧ C ⊆ A) → Fun F)
3		fdm 5226	. . . . . . 7 ⊢ (F:A–→B → dom F = A)
4	3	sseq2d 3299	. . . . . 6 ⊢ (F:A–→B → (C ⊆ dom F ↔ C ⊆ A))
5	4	biimpar 471	. . . . 5 ⊢ ((F:A–→B ∧ C ⊆ A) → C ⊆ dom F)
6		fores 5278	. . . . 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 5164	. . . 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 4792	. . 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 5292	. 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 3257   “ cima 4722  ^◡ccnv 4771  dom cdm 4772   ↾ cres 4774  Fun wfun 4775  –→wf 4777  –1-1→wf1 4778  –onto→wfo 4779  –1-1-onto→wf1o 4780

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794

This theorem is referenced by:  f1imacnv  5302  isoini2  5498  swapres  5512  xpassen  6057  enpw1pw  6075  ncdisjun  6136  sbthlem1  6203  lenc  6223