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Theorem fvelsetpreimafv 45653
Description: There is an element in a preimage 𝑆 of function values so that 𝑆 is the preimage of the function value at this element. (Contributed by AV, 8-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
fvelsetpreimafv ((𝐹 Fn 𝐴𝑆𝑃) → ∃𝑥𝑆 𝑆 = (𝐹 “ {(𝐹𝑥)}))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑆,𝑧
Allowed substitution hints:   𝑃(𝑥,𝑧)

Proof of Theorem fvelsetpreimafv
StepHypRef Expression
1 preimafvsnel 45645 . . . . . . 7 ((𝐹 Fn 𝐴𝑥𝐴) → 𝑥 ∈ (𝐹 “ {(𝐹𝑥)}))
21adantrr 716 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝑥𝐴𝑆 = (𝐹 “ {(𝐹𝑥)}))) → 𝑥 ∈ (𝐹 “ {(𝐹𝑥)}))
3 eleq2 2827 . . . . . . 7 (𝑆 = (𝐹 “ {(𝐹𝑥)}) → (𝑥𝑆𝑥 ∈ (𝐹 “ {(𝐹𝑥)})))
43ad2antll 728 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝑥𝐴𝑆 = (𝐹 “ {(𝐹𝑥)}))) → (𝑥𝑆𝑥 ∈ (𝐹 “ {(𝐹𝑥)})))
52, 4mpbird 257 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝑥𝐴𝑆 = (𝐹 “ {(𝐹𝑥)}))) → 𝑥𝑆)
6 simprr 772 . . . . 5 ((𝐹 Fn 𝐴 ∧ (𝑥𝐴𝑆 = (𝐹 “ {(𝐹𝑥)}))) → 𝑆 = (𝐹 “ {(𝐹𝑥)}))
75, 6jca 513 . . . 4 ((𝐹 Fn 𝐴 ∧ (𝑥𝐴𝑆 = (𝐹 “ {(𝐹𝑥)}))) → (𝑥𝑆𝑆 = (𝐹 “ {(𝐹𝑥)})))
87ex 414 . . 3 (𝐹 Fn 𝐴 → ((𝑥𝐴𝑆 = (𝐹 “ {(𝐹𝑥)})) → (𝑥𝑆𝑆 = (𝐹 “ {(𝐹𝑥)}))))
98reximdv2 3162 . 2 (𝐹 Fn 𝐴 → (∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}) → ∃𝑥𝑆 𝑆 = (𝐹 “ {(𝐹𝑥)})))
10 setpreimafvex.p . . 3 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1110elsetpreimafv 45651 . 2 (𝑆𝑃 → ∃𝑥𝐴 𝑆 = (𝐹 “ {(𝐹𝑥)}))
129, 11impel 507 1 ((𝐹 Fn 𝐴𝑆𝑃) → ∃𝑥𝑆 𝑆 = (𝐹 “ {(𝐹𝑥)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {cab 2714  wrex 3074  {csn 4591  ccnv 5637  cima 5641   Fn wfn 6496  cfv 6501
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-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-fv 6509
This theorem is referenced by:  imaelsetpreimafv  45661
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