Theorem funimacnv 6617

Description: The image of the preimage of a function. (Contributed by NM, 25-May-2004.)

Assertion

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

Proof of Theorem funimacnv

Step	Hyp	Ref	Expression
1		df-ima 5667	. . 3 ⊢ (𝐹 “ (◡𝐹 “ 𝐴)) = ran (𝐹 ↾ (◡𝐹 “ 𝐴))
2		funcnvres2 6616	. . . 4 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴)))
3	2	rneqd 5918	. . 3 ⊢ (Fun 𝐹 → ran ◡(◡𝐹 ↾ 𝐴) = ran (𝐹 ↾ (◡𝐹 “ 𝐴)))
4	1, 3	eqtr4id 2789	. 2 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = ran ◡(◡𝐹 ↾ 𝐴))
5		df-rn 5665	. . . 4 ⊢ ran 𝐹 = dom ◡𝐹
6	5	ineq2i 4192	. . 3 ⊢ (𝐴 ∩ ran 𝐹) = (𝐴 ∩ dom ◡𝐹)
7		dmres 5999	. . 3 ⊢ dom (◡𝐹 ↾ 𝐴) = (𝐴 ∩ dom ◡𝐹)
8		dfdm4 5875	. . 3 ⊢ dom (◡𝐹 ↾ 𝐴) = ran ◡(◡𝐹 ↾ 𝐴)
9	6, 7, 8	3eqtr2ri 2765	. 2 ⊢ ran ◡(◡𝐹 ↾ 𝐴) = (𝐴 ∩ ran 𝐹)
10	4, 9	eqtrdi 2786	1 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹))

Syntax hints:   → wi 4   = wceq 1540   ∩ cin 3925  ^◡ccnv 5653  dom cdm 5654  ran crn 5655   ↾ cres 5656   “ cima 5657  Fun wfun 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-fun 6533

This theorem is referenced by:  funimass1  6618  funimass2  6619  rescnvimafod  7063  isercolllem2  15682  isercolllem3  15683  isercoll  15684  cncls  23212  preimane  32648  fnpreimac  32649  ffsrn  32706  gsumhashmul  33055  zarcmplem  33912  cvmliftlem15  35320  fcoreslem2  47093  imaelsetpreimafv  47409