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Theorem intimafv 32439
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 6948 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
21inteqd 4948 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦})
3 fvex 6898 . . . . 5 (𝐹𝑥) ∈ V
43rgenw 3059 . . . 4 𝑥𝐴 (𝐹𝑥) ∈ V
5 iinabrex 32309 . . . 4 (∀𝑥𝐴 (𝐹𝑥) ∈ V → 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
64, 5ax-mp 5 . . 3 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
7 eqcom 2733 . . . . . 6 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
87rexbii 3088 . . . . 5 (∃𝑥𝐴 (𝐹𝑥) = 𝑦 ↔ ∃𝑥𝐴 𝑦 = (𝐹𝑥))
98abbii 2796 . . . 4 {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
109inteqi 4947 . . 3 {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
116, 10eqtr4i 2757 . 2 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 (𝐹𝑥) = 𝑦}
122, 11eqtr4di 2784 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = 𝑥𝐴 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {cab 2703  wral 3055  wrex 3064  Vcvv 3468  wss 3943   cint 4943   ciin 4991  dom cdm 5669  cima 5672  Fun wfun 6531  cfv 6537
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-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iin 4993  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-fv 6545
This theorem is referenced by:  zarclsint  33382  zarcmplem  33391
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