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Theorem fimacnvinrn2 7006
Description: Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 17-Feb-2017.)
Assertion
Ref Expression
fimacnvinrn2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹𝐴) = (𝐹 “ (𝐴𝐵)))

Proof of Theorem fimacnvinrn2
StepHypRef Expression
1 inass 4166 . . . 4 ((𝐴𝐵) ∩ ran 𝐹) = (𝐴 ∩ (𝐵 ∩ ran 𝐹))
2 sseqin2 4162 . . . . . . 7 (ran 𝐹𝐵 ↔ (𝐵 ∩ ran 𝐹) = ran 𝐹)
32biimpi 215 . . . . . 6 (ran 𝐹𝐵 → (𝐵 ∩ ran 𝐹) = ran 𝐹)
43adantl 482 . . . . 5 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐵 ∩ ran 𝐹) = ran 𝐹)
54ineq2d 4159 . . . 4 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐴 ∩ (𝐵 ∩ ran 𝐹)) = (𝐴 ∩ ran 𝐹))
61, 5eqtrid 2788 . . 3 ((Fun 𝐹 ∧ ran 𝐹𝐵) → ((𝐴𝐵) ∩ ran 𝐹) = (𝐴 ∩ ran 𝐹))
76imaeq2d 5999 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
8 fimacnvinrn 7005 . . 3 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)))
98adantr 481 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹 “ (𝐴𝐵)) = (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)))
10 fimacnvinrn 7005 . . 3 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
1110adantr 481 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
127, 9, 113eqtr4rd 2787 1 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹𝐴) = (𝐹 “ (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  cin 3897  wss 3898  ccnv 5619  ran crn 5621  cima 5623  Fun wfun 6473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-fun 6481  df-fn 6482  df-f 6483  df-fo 6485
This theorem is referenced by:  eulerpartgbij  32639  orvcval4  32727  preimaioomnf  44602
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