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Theorem intimafv 32720
Description: The intersection of an image set, as an indexed intersection of function values. (Contributed by Thierry Arnoux, 15-Jun-2024.)
Assertion
Ref Expression
intimafv ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 (𝐹𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem intimafv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfimafn 6971 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
21inteqd 4951 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
3 fvex 6919 . . . . 5 (𝐹𝑥) ∈ V
43rgenw 3065 . . . 4 𝑥𝐴 (𝐹𝑥) ∈ V
5 iinabrex 32582 . . . 4 (∀𝑥𝐴 (𝐹𝑥) ∈ V → 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
64, 5ax-mp 5 . . 3 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
7 eqcom 2744 . . . . . 6 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
87rexbii 3094 . . . . 5 (∃𝑥𝐴 (𝐹𝑥) = 𝑦 ↔ ∃𝑥𝐴 𝑦 = (𝐹𝑥))
98abbii 2809 . . . 4 {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
109inteqi 4950 . . 3 {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
116, 10eqtr4i 2768 . 2 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦}
122, 11eqtr4di 2795 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  wss 3951   cint 4946   ciin 4992  dom cdm 5685  cima 5688  Fun wfun 6555  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iin 4994  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  zarclsint  33871  zarcmplem  33880
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