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Theorem elsetpreimafvssdm 46354
Description: An element of the class 𝑃 of all preimages of function values is a subset of the domain of the function. (Contributed by AV, 8-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
elsetpreimafvssdm ((𝐹 Fn 𝐴𝑆𝑃) → 𝑆𝐴)
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑆,𝑧
Allowed substitution hints:   𝑃(𝑥,𝑧)

Proof of Theorem elsetpreimafvssdm
StepHypRef Expression
1 setpreimafvex.p . . . 4 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
21elsetpreimafv 46353 . . 3 (𝑆𝑃 → ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}))
3 cnvimass 6081 . . . . . . . . 9 (𝐹 “ {(𝐹𝑥)}) ⊆ dom 𝐹
4 fndm 6653 . . . . . . . . 9 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
53, 4sseqtrid 4035 . . . . . . . 8 (𝐹 Fn 𝐴 → (𝐹 “ {(𝐹𝑥)}) ⊆ 𝐴)
65adantr 479 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 “ {(𝐹𝑥)}) ⊆ 𝐴)
7 sseq1 4008 . . . . . . 7 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝑆𝐴 ↔ (𝐹 “ {(𝐹𝑥)}) ⊆ 𝐴))
86, 7syl5ibrcom 246 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑆 = (𝐹 “ {(𝐹𝑥)}) → 𝑆𝐴))
98expcom 412 . . . . 5 (𝑥𝐴 → (𝐹 Fn 𝐴 → (𝑆 = (𝐹 “ {(𝐹𝑥)}) → 𝑆𝐴)))
109com23 86 . . . 4 (𝑥𝐴 → (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝐹 Fn 𝐴𝑆𝐴)))
1110rexlimiv 3146 . . 3 (∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝐹 Fn 𝐴𝑆𝐴))
122, 11syl 17 . 2 (𝑆𝑃 → (𝐹 Fn 𝐴𝑆𝐴))
1312impcom 406 1 ((𝐹 Fn 𝐴𝑆𝑃) → 𝑆𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  {cab 2707  wrex 3068  wss 3949  {csn 4629  ccnv 5676  dom cdm 5677  cima 5680   Fn wfn 6539  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fn 6547
This theorem is referenced by:  preimafvsspwdm  46357  uniimaelsetpreimafv  46364
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