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Theorem uniimaelsetpreimafv 46054
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 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}
Assertion
Ref Expression
uniimaelsetpreimafv ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) β†’ βˆͺ (𝐹 β€œ 𝑆) ∈ ran 𝐹)
Distinct variable groups:   π‘₯,𝐴,𝑧   π‘₯,𝐹,𝑧   π‘₯,𝑆,𝑧   π‘₯,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 ∣ βˆƒπ‘₯ ∈ 𝐴 𝑧 = (◑𝐹 β€œ {(πΉβ€˜π‘₯)})}
210nelsetpreimafv 46048 . . . 4 (𝐹 Fn 𝐴 β†’ βˆ… βˆ‰ 𝑃)
3 elnelne2 3058 . . . . . 6 ((𝑆 ∈ 𝑃 ∧ βˆ… βˆ‰ 𝑃) β†’ 𝑆 β‰  βˆ…)
4 n0 4346 . . . . . 6 (𝑆 β‰  βˆ… ↔ βˆƒπ‘¦ 𝑦 ∈ 𝑆)
53, 4sylib 217 . . . . 5 ((𝑆 ∈ 𝑃 ∧ βˆ… βˆ‰ 𝑃) β†’ βˆƒπ‘¦ 𝑦 ∈ 𝑆)
65expcom 414 . . . 4 (βˆ… βˆ‰ 𝑃 β†’ (𝑆 ∈ 𝑃 β†’ βˆƒπ‘¦ 𝑦 ∈ 𝑆))
72, 6syl 17 . . 3 (𝐹 Fn 𝐴 β†’ (𝑆 ∈ 𝑃 β†’ βˆƒπ‘¦ 𝑦 ∈ 𝑆))
87imp 407 . 2 ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) β†’ βˆƒπ‘¦ 𝑦 ∈ 𝑆)
91imaelsetpreimafv 46053 . . . . . 6 ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ∧ 𝑦 ∈ 𝑆) β†’ (𝐹 β€œ 𝑆) = {(πΉβ€˜π‘¦)})
1093expa 1118 . . . . 5 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ (𝐹 β€œ 𝑆) = {(πΉβ€˜π‘¦)})
1110unieqd 4922 . . . 4 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ βˆͺ (𝐹 β€œ 𝑆) = βˆͺ {(πΉβ€˜π‘¦)})
12 fvex 6904 . . . . 5 (πΉβ€˜π‘¦) ∈ V
1312unisn 4930 . . . 4 βˆͺ {(πΉβ€˜π‘¦)} = (πΉβ€˜π‘¦)
1411, 13eqtrdi 2788 . . 3 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ βˆͺ (𝐹 β€œ 𝑆) = (πΉβ€˜π‘¦))
15 dffn3 6730 . . . . . 6 (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟢ran 𝐹)
1615biimpi 215 . . . . 5 (𝐹 Fn 𝐴 β†’ 𝐹:𝐴⟢ran 𝐹)
1716ad2antrr 724 . . . 4 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ 𝐹:𝐴⟢ran 𝐹)
181elsetpreimafvssdm 46044 . . . . 5 ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) β†’ 𝑆 βŠ† 𝐴)
1918sselda 3982 . . . 4 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ 𝑦 ∈ 𝐴)
2017, 19ffvelcdmd 7087 . . 3 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ (πΉβ€˜π‘¦) ∈ ran 𝐹)
2114, 20eqeltrd 2833 . 2 (((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) ∧ 𝑦 ∈ 𝑆) β†’ βˆͺ (𝐹 β€œ 𝑆) ∈ ran 𝐹)
228, 21exlimddv 1938 1 ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃) β†’ βˆͺ (𝐹 β€œ 𝑆) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {cab 2709   β‰  wne 2940   βˆ‰ wnel 3046  βˆƒwrex 3070  βˆ…c0 4322  {csn 4628  βˆͺ cuni 4908  β—‘ccnv 5675  ran crn 5677   β€œ cima 5679   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543
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 5299  ax-nul 5306  ax-pr 5427
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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551
This theorem is referenced by:  imasetpreimafvbijlemf  46059  fundcmpsurbijinjpreimafv  46065
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