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Theorem fimacnvinrn2 6672
 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 4086 . . . 4 ((𝐴𝐵) ∩ ran 𝐹) = (𝐴 ∩ (𝐵 ∩ ran 𝐹))
2 sseqin2 4082 . . . . . . 7 (ran 𝐹𝐵 ↔ (𝐵 ∩ ran 𝐹) = ran 𝐹)
32biimpi 208 . . . . . 6 (ran 𝐹𝐵 → (𝐵 ∩ ran 𝐹) = ran 𝐹)
43adantl 474 . . . . 5 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐵 ∩ ran 𝐹) = ran 𝐹)
54ineq2d 4079 . . . 4 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐴 ∩ (𝐵 ∩ ran 𝐹)) = (𝐴 ∩ ran 𝐹))
61, 5syl5eq 2828 . . 3 ((Fun 𝐹 ∧ ran 𝐹𝐵) → ((𝐴𝐵) ∩ ran 𝐹) = (𝐴 ∩ ran 𝐹))
76imaeq2d 5775 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
8 fimacnvinrn 6671 . . 3 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)))
98adantr 473 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹 “ (𝐴𝐵)) = (𝐹 “ ((𝐴𝐵) ∩ ran 𝐹)))
10 fimacnvinrn 6671 . . 3 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
1110adantr 473 . 2 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
127, 9, 113eqtr4rd 2827 1 ((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹𝐴) = (𝐹 “ (𝐴𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∩ cin 3830   ⊆ wss 3831  ◡ccnv 5410  ran crn 5412   “ cima 5414  Fun wfun 6187 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-sep 5064  ax-nul 5071  ax-pr 5190 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3419  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4182  df-if 4354  df-sn 4445  df-pr 4447  df-op 4451  df-uni 4718  df-br 4935  df-opab 4997  df-id 5316  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-fo 6199  df-fv 6201 This theorem is referenced by:  eulerpartgbij  31307  orvcval4  31396  preimaioomnf  42463
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