Theorem funimacnv 5207

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 4560	. . 3 ⊢ (𝐹 “ (◡𝐹 “ 𝐴)) = ran (𝐹 ↾ (◡𝐹 “ 𝐴))
2		funcnvres2 5206	. . . 4 ⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴)))
3	2	rneqd 4776	. . 3 ⊢ (Fun 𝐹 → ran ◡(◡𝐹 ↾ 𝐴) = ran (𝐹 ↾ (◡𝐹 “ 𝐴)))
4	1, 3	eqtr4id 2192	. 2 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = ran ◡(◡𝐹 ↾ 𝐴))
5		df-rn 4558	. . . 4 ⊢ ran 𝐹 = dom ◡𝐹
6	5	ineq2i 3279	. . 3 ⊢ (𝐴 ∩ ran 𝐹) = (𝐴 ∩ dom ◡𝐹)
7		dmres 4848	. . 3 ⊢ dom (◡𝐹 ↾ 𝐴) = (𝐴 ∩ dom ◡𝐹)
8		dfdm4 4739	. . 3 ⊢ dom (◡𝐹 ↾ 𝐴) = ran ◡(◡𝐹 ↾ 𝐴)
9	6, 7, 8	3eqtr2ri 2168	. 2 ⊢ ran ◡(◡𝐹 ↾ 𝐴) = (𝐴 ∩ ran 𝐹)
10	4, 9	eqtrdi 2189	1 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹))

Syntax hints:   → wi 4   = wceq 1332   ∩ cin 3075  ^◡ccnv 4546  dom cdm 4547  ran crn 4548   ↾ cres 4549   “ cima 4550  Fun wfun 5125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-fun 5133

This theorem is referenced by:  funimass1  5208  funimass2  5209