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Theorem elsetpreimafvssdm 46054
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 46053 . . 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 482 . . . . . . 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 415 . . . . 5 (𝑥𝐴 → (𝐹 Fn 𝐴 → (𝑆 = (𝐹 “ {(𝐹𝑥)}) → 𝑆𝐴)))
109com23 86 . . . 4 (𝑥𝐴 → (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝐹 Fn 𝐴𝑆𝐴)))
1110rexlimiv 3149 . . 3 (∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝐹 Fn 𝐴𝑆𝐴))
122, 11syl 17 . 2 (𝑆𝑃 → (𝐹 Fn 𝐴𝑆𝐴))
1312impcom 409 1 ((𝐹 Fn 𝐴𝑆𝑃) → 𝑆𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  wrex 3071  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 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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  46057  uniimaelsetpreimafv  46064
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