Theorem fimacnvinrn 6817

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 6811	. 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)))
2		funforn 6572	. . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹–onto→ran 𝐹)
3		fof 6565	. . . . 5 ⊢ (𝐹:dom 𝐹–onto→ran 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
4	2, 3	sylbi 220	. . . 4 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
5		fimacnv 6816	. . . 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 4139	. 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹))
8		cnvresima 6054	. . 3 ⊢ (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹)
9		resdm2 6055	. . . . . 6 ⊢ (𝐹 ↾ dom 𝐹) = ◡◡𝐹
10		funrel 6341	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
11		dfrel2 6013	. . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
12	10, 11	sylib 221	. . . . . 6 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹)
13	9, 12	syl5eq 2845	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
14	13	cnveqd 5710	. . . 4 ⊢ (Fun 𝐹 → ◡(𝐹 ↾ dom 𝐹) = ◡𝐹)
15	14	imaeq1d 5895	. . 3 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = (◡𝐹 “ 𝐴))
16	8, 15	syl5eqr 2847	. 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ dom 𝐹) = (◡𝐹 “ 𝐴))
17	1, 7, 16	3eqtrrd 2838	1 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))

Syntax hints:   → wi 4   = wceq 1538   ∩ cin 3880  ^◡ccnv 5518  dom cdm 5519  ran crn 5520   ↾ cres 5521   “ cima 5522  Rel wrel 5524  Fun wfun 6318  ⟶wf 6320  –onto→wfo 6322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332

This theorem is referenced by:  fimacnvinrn2  6818  preiman0  30469