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

Theorem elsetpreimafvssdm 44299
 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 44298 . . 3 (𝑆𝑃 → ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}))
3 cnvimass 5925 . . . . . . . . 9 (𝐹 “ {(𝐹𝑥)}) ⊆ dom 𝐹
4 fndm 6440 . . . . . . . . 9 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
53, 4sseqtrid 3946 . . . . . . . 8 (𝐹 Fn 𝐴 → (𝐹 “ {(𝐹𝑥)}) ⊆ 𝐴)
65adantr 484 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹 “ {(𝐹𝑥)}) ⊆ 𝐴)
7 sseq1 3919 . . . . . . 7 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝑆𝐴 ↔ (𝐹 “ {(𝐹𝑥)}) ⊆ 𝐴))
86, 7syl5ibrcom 250 . . . . . 6 ((𝐹 Fn 𝐴𝑥𝐴) → (𝑆 = (𝐹 “ {(𝐹𝑥)}) → 𝑆𝐴))
98expcom 417 . . . . 5 (𝑥𝐴 → (𝐹 Fn 𝐴 → (𝑆 = (𝐹 “ {(𝐹𝑥)}) → 𝑆𝐴)))
109com23 86 . . . 4 (𝑥𝐴 → (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝐹 Fn 𝐴𝑆𝐴)))
1110rexlimiv 3204 . . 3 (∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝐹 Fn 𝐴𝑆𝐴))
122, 11syl 17 . 2 (𝑆𝑃 → (𝐹 Fn 𝐴𝑆𝐴))
1312impcom 411 1 ((𝐹 Fn 𝐴𝑆𝑃) → 𝑆𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∃wrex 3071   ⊆ wss 3860  {csn 4525  ◡ccnv 5526  dom cdm 5527   “ cima 5530   Fn wfn 6334  ‘cfv 6339 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-xp 5533  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-fn 6342 This theorem is referenced by:  preimafvsspwdm  44302  uniimaelsetpreimafv  44309
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