Theorem fimacnvinrn 7105

Description: Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 21-Jan-2017.)

Assertion

Ref	Expression
fimacnvinrn	⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))

Proof of Theorem fimacnvinrn

Step	Hyp	Ref	Expression
1		inpreima 7097	. 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)))
2		funforn 6841	. . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹–onto→ran 𝐹)
3		fof 6834	. . . . 5 ⊢ (𝐹:dom 𝐹–onto→ran 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
4	2, 3	sylbi 217	. . . 4 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
5		fimacnv 6769	. . . 4 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 → (◡𝐹 “ ran 𝐹) = dom 𝐹)
6	4, 5	syl 17	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ ran 𝐹) = dom 𝐹)
7	6	ineq2d 4241	. 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹))
8		cnvresima 6261	. . 3 ⊢ (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹)
9		resdm2 6262	. . . . . 6 ⊢ (𝐹 ↾ dom 𝐹) = ◡◡𝐹
10		funrel 6595	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
11		dfrel2 6220	. . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
12	10, 11	sylib 218	. . . . . 6 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹)
13	9, 12	eqtrid 2792	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
14	13	cnveqd 5900	. . . 4 ⊢ (Fun 𝐹 → ◡(𝐹 ↾ dom 𝐹) = ◡𝐹)
15	14	imaeq1d 6088	. . 3 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = (◡𝐹 “ 𝐴))
16	8, 15	eqtr3id 2794	. 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ dom 𝐹) = (◡𝐹 “ 𝐴))
17	1, 7, 16	3eqtrrd 2785	1 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))

Syntax hints:   → wi 4   = wceq 1537   ∩ cin 3975  ^◡ccnv 5699  dom cdm 5700  ran crn 5701   ↾ cres 5702   “ cima 5703  Rel wrel 5705  Fun wfun 6567  ⟶wf 6569  –onto→wfo 6571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579

This theorem is referenced by:  fimacnvinrn2  7106  preiman0  32721