Theorem fimacnvinrn 7018

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 7011	. 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)))
2		funforn 6754	. . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹–onto→ran 𝐹)
3		fof 6747	. . . . 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 6685	. . . 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 4173	. 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹))
8		cnvresima 6189	. . 3 ⊢ (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹)
9		resdm2 6190	. . . . . 6 ⊢ (𝐹 ↾ dom 𝐹) = ◡◡𝐹
10		funrel 6510	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
11		dfrel2 6148	. . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
12	10, 11	sylib 218	. . . . . 6 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹)
13	9, 12	eqtrid 2784	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
14	13	cnveqd 5825	. . . 4 ⊢ (Fun 𝐹 → ◡(𝐹 ↾ dom 𝐹) = ◡𝐹)
15	14	imaeq1d 6019	. . 3 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = (◡𝐹 “ 𝐴))
16	8, 15	eqtr3id 2786	. 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ dom 𝐹) = (◡𝐹 “ 𝐴))
17	1, 7, 16	3eqtrrd 2777	1 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))

Syntax hints:   → wi 4   = wceq 1542   ∩ cin 3901  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628  Rel wrel 5630  Fun wfun 6487  ⟶wf 6489  –onto→wfo 6491

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499

This theorem is referenced by:  fimacnvinrn2  7019  preiman0  32792  psrbasfsupp  33697