Theorem fimacnvinrn 7092

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 7084	. 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)))
2		funforn 6822	. . . . 5 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ontoran 𝐹)
3		fof 6815	. . . . 5 ⊢ (𝐹:dom 𝐹⟶ontoran 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
4	2, 3	sylbi 220	. . . 4 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
5		fimacnv 6751	. . . 4 ⊢ (𝐹:dom 𝐹⟶ran 𝐹 → (◡𝐹 “ ran 𝐹) = dom 𝐹)
6	4, 5	syl 18	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ ran 𝐹) = dom 𝐹)
7	6	ineq2d 4120	. 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹))
8		cnvresima 6243	. . 3 ⊢ (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ dom 𝐹)
9		resdm2 6244	. . . . . 6 ⊢ (𝐹 ↾ dom 𝐹) = ◡◡𝐹
10		funrel 6573	. . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹)
11		dfrel2 6192	. . . . . . 7 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
12	10, 11	sylib 221	. . . . . 6 ⊢ (Fun 𝐹 → ◡◡𝐹 = 𝐹)
13	9, 12	eqtrid 2810	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
14	13	cnveqd 5859	. . . 4 ⊢ (Fun 𝐹 → ◡(𝐹 ↾ dom 𝐹) = ◡𝐹)
15	14	imaeq1d 6062	. . 3 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ dom 𝐹) “ 𝐴) = (◡𝐹 “ 𝐴))
16	8, 15	eqtr3id 2812	. 2 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) ∩ dom 𝐹) = (◡𝐹 “ 𝐴))
17	1, 7, 16	3eqtrrd 2803	1 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∩ cin 4092  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ↾ cres 5657   “ cima 5658  Rel wrel 5660  Fun wfun 6549  ⟶wf 6551  ⟶ontowfo 6553

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6557  df-fn 6558  df-f 6559  df-fo 6561

This theorem is used by:  fimacnvinrn2  7093  preiman0  33371  psrbasfsupp  34210