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Theorem uniimaelsetpreimafv 45900
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 45894 . . . 4 (𝐹 Fn 𝐴 → ∅ ∉ 𝑃)
3 elnelne2 3058 . . . . . 6 ((𝑆𝑃 ∧ ∅ ∉ 𝑃) → 𝑆 ≠ ∅)
4 n0 4343 . . . . . 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 45899 . . . . . 6 ((𝐹 Fn 𝐴𝑆𝑃𝑦𝑆) → (𝐹𝑆) = {(𝐹𝑦)})
1093expa 1118 . . . . 5 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑆) = {(𝐹𝑦)})
1110unieqd 4916 . . . 4 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑆) = {(𝐹𝑦)})
12 fvex 6892 . . . . 5 (𝐹𝑦) ∈ V
1312unisn 4924 . . . 4 {(𝐹𝑦)} = (𝐹𝑦)
1411, 13eqtrdi 2788 . . 3 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑆) = (𝐹𝑦))
15 dffn3 6718 . . . . . 6 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
1615biimpi 215 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
1716ad2antrr 724 . . . 4 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → 𝐹:𝐴⟶ran 𝐹)
181elsetpreimafvssdm 45890 . . . . 5 ((𝐹 Fn 𝐴𝑆𝑃) → 𝑆𝐴)
1918sselda 3979 . . . 4 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → 𝑦𝐴)
2017, 19ffvelcdmd 7073 . . 3 (((𝐹 Fn 𝐴𝑆𝑃) ∧ 𝑦𝑆) → (𝐹𝑦) ∈ ran 𝐹)
2114, 20eqeltrd 2833 . 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 1541  wex 1781  wcel 2106  {cab 2709  wne 2940  wnel 3046  wrex 3070  c0 4319  {csn 4623   cuni 4902  ccnv 5669  ran crn 5671  cima 5673   Fn wfn 6528  wf 6529  cfv 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541
This theorem is referenced by:  imasetpreimafvbijlemf  45905  fundcmpsurbijinjpreimafv  45911
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