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Theorem fimacnvinrn2 6512
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 3966 . . . 4 ((𝐴𝐵) ∩ ran 𝐹) = (𝐴 ∩ (𝐵 ∩ ran 𝐹))
2 sseqin2 3960 . . . . . . 7 (ran 𝐹𝐵 ↔ (𝐵 ∩ ran 𝐹) = ran 𝐹)
32biimpi 206 . . . . . 6 (ran 𝐹𝐵 → (𝐵 ∩ ran 𝐹) = ran 𝐹)
43adantl 473 . . . . 5 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐵 ∩ ran 𝐹) = ran 𝐹)
54ineq2d 3957 . . . 4 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐴 ∩ (𝐵 ∩ ran 𝐹)) = (𝐴 ∩ ran 𝐹))
61, 5syl5eq 2806 . . 3 ((Fun 𝐹 ∧ ran 𝐹𝐵) → ((𝐴𝐵) ∩ ran 𝐹) = (𝐴 ∩ ran 𝐹))
76imaeq2d 5624 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
8 fimacnvinrn 6511 . . 3 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)))
98adantr 472 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹 “ (𝐴𝐵)) = (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)))
10 fimacnvinrn 6511 . . 3 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
1110adantr 472 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
127, 9, 113eqtr4rd 2805 1 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹𝐴) = (𝐹 “ (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  cin 3714  wss 3715  ccnv 5265  ran crn 5267  cima 5269  Fun wfun 6043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057
This theorem is referenced by:  eulerpartgbij  30743  orvcval4  30831  preimaioomnf  41435
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