Theorem f1ores 2813

Description: The restriction of a one-to-one function maps one-to-one onto the image.

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		fores 2794	. . . . 5 ⊢ ((Fun F ∧ C ⊆ dom F) → (F ↾ C):C–onto→(F “ C))
2		ffun 2754	. . . . . 6 ⊢ (F:A–→B → Fun F)
3	2	adantr 306	. . . . 5 ⊢ ((F:A–→B ∧ C ⊆ A) → Fun F)
4		fdm 2756	. . . . . . 7 ⊢ (F:A–→B → dom F = A)
5	4	sseq2d 1528	. . . . . 6 ⊢ (F:A–→B → (C ⊆ dom F ↔ C ⊆ A))
6	5	biimpar 325	. . . . 5 ⊢ ((F:A–→B ∧ C ⊆ A) → C ⊆ dom F)
7	1, 3, 6	sylanc 361	. . . 4 ⊢ ((F:A–→B ∧ C ⊆ A) → (F ↾ C):C–onto→(F “ C))
8		funres11 2709	. . . 4 ⊢ (Fun ◡F → Fun ◡(F ↾ C))
9	7, 8	anim12i 268	. . 3 ⊢ (((F:A–→B ∧ C ⊆ A) ∧ Fun ◡F) → ((F ↾ C):C–onto→(F “ C) ∧ Fun ◡(F ↾ C)))
10	9	an1rs 373	. 2 ⊢ (((F:A–→B ∧ Fun ◡F) ∧ C ⊆ A) → ((F ↾ C):C–onto→(F “ C) ∧ Fun ◡(F ↾ C)))
11		df-f1 2435	. . 3 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F))
12	11	anbi1i 368	. 2 ⊢ ((F:A–1-1→B ∧ C ⊆ A) ↔ ((F:A–→B ∧ Fun ◡F) ∧ C ⊆ A))
13		f1o3 2805	. 2 ⊢ ((F ↾ C):C–1-1-onto→(F “ C) ↔ ((F ↾ C):C–onto→(F “ C) ∧ Fun ◡(F ↾ C)))
14	10, 12, 13	3imtr4 192	1 ⊢ ((F:A–1-1→B ∧ C ⊆ A) → (F ↾ C):C–1-1-onto→(F “ C))

Syntax hints:   → wi 2   ∧ wa 196   ⊆ wss 1487  ^◡ccnv 2409  dom cdm 2410   ↾ cres 2412   “ cima 2413  Fun wfun 2416  –→wf 2418  –1-1→wf1 2419  –onto→wfo 2420  –1-1-onto→wf1o 2421

This theorem is referenced by:  f1imacnv 2814  f1imaen 3327  phplem5 3407  php3 3411  ssfi 3430  fiint 3445

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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