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Theorem intimafv 32726
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 4956 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
3 fvex 6920 . . . . 5 (𝐹𝑥) ∈ V
43rgenw 3063 . . . 4 𝑥𝐴 (𝐹𝑥) ∈ V
5 iinabrex 32589 . . . 4 (∀𝑥𝐴 (𝐹𝑥) ∈ V → 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
64, 5ax-mp 5 . . 3 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
7 eqcom 2742 . . . . . 6 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
87rexbii 3092 . . . . 5 (∃𝑥𝐴 (𝐹𝑥) = 𝑦 ↔ ∃𝑥𝐴 𝑦 = (𝐹𝑥))
98abbii 2807 . . . 4 {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
109inteqi 4955 . . 3 {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
116, 10eqtr4i 2766 . 2 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦}
122, 11eqtr4di 2793 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  wss 3963   cint 4951   ciin 4997  dom cdm 5689  cima 5692  Fun wfun 6557  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iin 4999  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  zarclsint  33833  zarcmplem  33842
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