Theorem fimacnvinrn 7016

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 7009	. 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)))
2		funforn 6750	. . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹–onto→ran 𝐹)
3		fof 6743	. . . . 5 ⊢ (𝐹:dom 𝐹–onto→ran 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
4	2, 3	sylbi 219	. . . 4 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
5		fimacnv 6681	. . . 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 4152	. 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹))
8		cnvresima 6185	. . 3 ⊢ (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹)
9		resdm2 6186	. . . . . 6 ⊢ (𝐹 ↾ dom 𝐹) = ◡◡𝐹
10		funrel 6506	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
11		dfrel2 6144	. . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
12	10, 11	sylib 220	. . . . . 6 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹)
13	9, 12	eqtrid 2788	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
14	13	cnveqd 5820	. . . 4 ⊢ (Fun 𝐹 → ◡(𝐹 ↾ dom 𝐹) = ◡𝐹)
15	14	imaeq1d 6018	. . 3 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = (◡𝐹 “ 𝐴))
16	8, 15	eqtr3id 2790	. 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ dom 𝐹) = (◡𝐹 “ 𝐴))
17	1, 7, 16	3eqtrrd 2781	1 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))

Syntax hints:   → wi 4   = wceq 1548   ∩ cin 3884  ^◡ccnv 5620  dom cdm 5621  ran crn 5622   ↾ cres 5623   “ cima 5624  Rel wrel 5626  Fun wfun 6483  ⟶wf 6485  –onto→wfo 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-fun 6491  df-fn 6492  df-f 6493  df-fo 6495

This theorem is referenced by:  fimacnvinrn2  7017  preiman0  32806  psrbasfsupp  33707