Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  uniimaelsetpreimafv Structured version   Visualization version   GIF version

Theorem uniimaelsetpreimafv 44848
Description: The union of the image of an element of the preimage of a function value is an element of the range of the function. (Contributed by AV, 5-Mar-2024.) (Revised by AV, 22-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
uniimaelsetpreimafv ((𝐹 Fn 𝐴𝑆𝑃) → (𝐹𝑆) ∈ ran 𝐹)
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑆,𝑧   𝑥,𝑃
Allowed substitution hint:   𝑃(𝑧)

Proof of Theorem uniimaelsetpreimafv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 setpreimafvex.p . . . . 5 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
210nelsetpreimafv 44842 . . . 4 (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
3 elnelne2 3060 . . . . . 6 ((𝑆𝑃 ∧ ∅ ∉ 𝑃) → 𝑆 ≠ ∅)
4 n0 4280 . . . . . 6 (𝑆 ≠ ∅ ↔ ∃𝑦 𝑦𝑆)
53, 4sylib 217 . . . . 5 ((𝑆𝑃 ∧ ∅ ∉ 𝑃) → ∃𝑦 𝑦𝑆)
65expcom 414 . . . 4 (∅ ∉ 𝑃 → (𝑆𝑃 → ∃𝑦 𝑦𝑆))
72, 6syl 17 . . 3 (𝐹 Fn 𝐴 → (𝑆𝑃 → ∃𝑦 𝑦𝑆))
87imp 407 . 2 ((𝐹 Fn 𝐴𝑆𝑃) → ∃𝑦 𝑦𝑆)
91imaelsetpreimafv 44847 . . . . . 6 ((𝐹 Fn 𝐴𝑆𝑃𝑦𝑆) → (𝐹𝑆) = {(𝐹𝑦)})
1093expa 1117 . . . . 5 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑆) = {(𝐹𝑦)})
1110unieqd 4853 . . . 4 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑆) = {(𝐹𝑦)})
12 fvex 6787 . . . . 5 (𝐹𝑦) ∈ V
1312unisn 4861 . . . 4 {(𝐹𝑦)} = (𝐹𝑦)
1411, 13eqtrdi 2794 . . 3 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑆) = (𝐹𝑦))
15 dffn3 6613 . . . . . 6 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
1615biimpi 215 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
1716ad2antrr 723 . . . 4 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → 𝐹:𝐴⟶ran 𝐹)
181elsetpreimafvssdm 44838 . . . . 5 ((𝐹 Fn 𝐴𝑆𝑃) → 𝑆𝐴)
1918sselda 3921 . . . 4 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → 𝑦𝐴)
2017, 19ffvelrnd 6962 . . 3 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑦) ∈ ran 𝐹)
2114, 20eqeltrd 2839 . 2 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑆) ∈ ran 𝐹)
228, 21exlimddv 1938 1 ((𝐹 Fn 𝐴𝑆𝑃) → (𝐹𝑆) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wne 2943  wnel 3049  wrex 3065  c0 4256  {csn 4561   cuni 4839  ccnv 5588  ran crn 5590  cima 5592   Fn wfn 6428  wf 6429  cfv 6433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441
This theorem is referenced by:  imasetpreimafvbijlemf  44853  fundcmpsurbijinjpreimafv  44859
  Copyright terms: Public domain W3C validator