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Theorem elsetpreimafveqfv 47317
Description: The elements of the preimage of a function value have the same function values. (Contributed by AV, 5-Mar-2024.)
Hypothesis
Ref Expression
setpreimafvex.p 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
Assertion
Ref Expression
elsetpreimafveqfv ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑋𝑆𝑌𝑆)) → (𝐹𝑋) = (𝐹𝑌))
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧   𝑥,𝑆,𝑧   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝑃(𝑥,𝑧)   𝑋(𝑧)   𝑌(𝑧)

Proof of Theorem elsetpreimafveqfv
StepHypRef Expression
1 setpreimafvex.p . . . . 5 𝑃 = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
21elsetpreimafvbi 47316 . . . 4 ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → (𝑌𝑆 ↔ (𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋))))
3 simpr 484 . . . . 5 ((𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)) → (𝐹𝑌) = (𝐹𝑋))
43eqcomd 2741 . . . 4 ((𝑌𝐴 ∧ (𝐹𝑌) = (𝐹𝑋)) → (𝐹𝑋) = (𝐹𝑌))
52, 4biimtrdi 253 . . 3 ((𝐹 Fn 𝐴𝑆𝑃𝑋𝑆) → (𝑌𝑆 → (𝐹𝑋) = (𝐹𝑌)))
653exp 1118 . 2 (𝐹 Fn 𝐴 → (𝑆𝑃 → (𝑋𝑆 → (𝑌𝑆 → (𝐹𝑋) = (𝐹𝑌)))))
763imp2 1348 1 ((𝐹 Fn 𝐴 ∧ (𝑆𝑃𝑋𝑆𝑌𝑆)) → (𝐹𝑋) = (𝐹𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  {csn 4631  ccnv 5688  cima 5692   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  imasetpreimafvbijlemfv  47327
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