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Theorem fimacnvinrn2 6257
 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 3785 . . . 4 ((𝐴𝐵) ∩ ran 𝐹) = (𝐴 ∩ (𝐵 ∩ ran 𝐹))
2 sseqin2 3779 . . . . . . 7 (ran 𝐹𝐵 ↔ (𝐵 ∩ ran 𝐹) = ran 𝐹)
32biimpi 205 . . . . . 6 (ran 𝐹𝐵 → (𝐵 ∩ ran 𝐹) = ran 𝐹)
43adantl 481 . . . . 5 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐵 ∩ ran 𝐹) = ran 𝐹)
54ineq2d 3776 . . . 4 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐴 ∩ (𝐵 ∩ ran 𝐹)) = (𝐴 ∩ ran 𝐹))
61, 5syl5eq 2656 . . 3 ((Fun 𝐹 ∧ ran 𝐹𝐵) → ((𝐴𝐵) ∩ ran 𝐹) = (𝐴 ∩ ran 𝐹))
76imaeq2d 5385 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
8 fimacnvinrn 6256 . . 3 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)))
98adantr 480 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹 “ (𝐴𝐵)) = (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)))
10 fimacnvinrn 6256 . . 3 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
1110adantr 480 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
127, 9, 113eqtr4rd 2655 1 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹𝐴) = (𝐹 “ (𝐴𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∩ cin 3539   ⊆ wss 3540  ◡ccnv 5037  ran crn 5039   “ cima 5041  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812 This theorem is referenced by:  eulerpartgbij  29761  orvcval4  29849  preimaioomnf  39606
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