Theorem fores 6802

Description: Restriction of an onto function. (Contributed by NM, 4-Mar-1997.)

Assertion

Ref	Expression
fores	⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))

Proof of Theorem fores

Step	Hyp	Ref	Expression
1		funres 6578	. . 3 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
2	1	anim1i 626	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (Fun (𝐹 ↾ 𝐴) ∧ 𝐴 ⊆ dom 𝐹))
3		df-fn 6539	. . 3 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ (Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴))
4		df-ima 5674	. . . . 5 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
5	4	eqcomi 2772	. . . 4 ⊢ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴)
6		df-fo 6542	. . . 4 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴)))
7	5, 6	mpbiran2 722	. . 3 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ (𝐹 ↾ 𝐴) Fn 𝐴)
8		ssdmres 6012	. . . 4 ⊢ (𝐴 ⊆ dom 𝐹 ↔ dom (𝐹 ↾ 𝐴) = 𝐴)
9	8	anbi2i 634	. . 3 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ 𝐴 ⊆ dom 𝐹) ↔ (Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴))
10	3, 7, 9	3bitr4i 306	. 2 ⊢ ((𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴) ↔ (Fun (𝐹 ↾ 𝐴) ∧ 𝐴 ⊆ dom 𝐹))
11	2, 10	sylibr 237	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ⊆ wss 3905  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664  Fun wfun 6530   Fn wfn 6531  –onto→wfo 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-res 5673  df-ima 5674  df-fun 6538  df-fn 6539  df-fo 6542

This theorem is referenced by:  fimadmfoALT  6803  resdif  6842  f1oweALT  7965  imafi  9271  f1opwfi  9309  fodomfi2  10040  fin1a2lem7  10385  znnen  16263  connima  23582  1stcfb  23602  1stckgenlem  23710  qtoprest  23874  re2ndc  24958  uniiccdif  25737  opnmblALT  25762  mbfimaopnlem  25814  ffsrn  33073  cycpmconjvlem  33461  erdszelem2  35684  ivthALT  36866  poimirlem26  38317  poimirlem27  38318  lmhmfgima  43831