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Theorem fimacnvinrn2 7064
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 4211 . . . 4 ((𝐴𝐵) ∩ ran 𝐹) = (𝐴 ∩ (𝐵 ∩ ran 𝐹))
2 sseqin2 4207 . . . . . . 7 (ran 𝐹𝐵 ↔ (𝐵 ∩ ran 𝐹) = ran 𝐹)
32biimpi 215 . . . . . 6 (ran 𝐹𝐵 → (𝐵 ∩ ran 𝐹) = ran 𝐹)
43adantl 481 . . . . 5 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐵 ∩ ran 𝐹) = ran 𝐹)
54ineq2d 4204 . . . 4 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐴 ∩ (𝐵 ∩ ran 𝐹)) = (𝐴 ∩ ran 𝐹))
61, 5eqtrid 2776 . . 3 ((Fun 𝐹 ∧ ran 𝐹𝐵) → ((𝐴𝐵) ∩ ran 𝐹) = (𝐴 ∩ ran 𝐹))
76imaeq2d 6049 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
8 fimacnvinrn 7063 . . 3 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)))
98adantr 480 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹 “ (𝐴𝐵)) = (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)))
10 fimacnvinrn 7063 . . 3 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
1110adantr 480 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
127, 9, 113eqtr4rd 2775 1 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹𝐴) = (𝐹 “ (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  cin 3939  wss 3940  ccnv 5665  ran crn 5667  cima 5669  Fun wfun 6527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539
This theorem is referenced by:  eulerpartgbij  33860  orvcval4  33948  preimaioomnf  45920
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