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Theorem preimane 32687
Description: Different elements have different preimages. (Contributed by Thierry Arnoux, 7-May-2023.)
Hypotheses
Ref Expression
preimane.f (𝜑 → Fun 𝐹)
preimane.x (𝜑𝑋𝑌)
preimane.y (𝜑𝑋 ∈ ran 𝐹)
preimane.1 (𝜑𝑌 ∈ ran 𝐹)
Assertion
Ref Expression
preimane (𝜑 → (𝐹 “ {𝑋}) ≠ (𝐹 “ {𝑌}))

Proof of Theorem preimane
StepHypRef Expression
1 preimane.x . . . 4 (𝜑𝑋𝑌)
2 preimane.y . . . . . 6 (𝜑𝑋 ∈ ran 𝐹)
3 sneqrg 4844 . . . . . 6 (𝑋 ∈ ran 𝐹 → ({𝑋} = {𝑌} → 𝑋 = 𝑌))
42, 3syl 17 . . . . 5 (𝜑 → ({𝑋} = {𝑌} → 𝑋 = 𝑌))
54necon3d 2959 . . . 4 (𝜑 → (𝑋𝑌 → {𝑋} ≠ {𝑌}))
61, 5mpd 15 . . 3 (𝜑 → {𝑋} ≠ {𝑌})
7 preimane.f . . . . 5 (𝜑 → Fun 𝐹)
8 funimacnv 6649 . . . . 5 (Fun 𝐹 → (𝐹 “ (𝐹 “ {𝑋})) = ({𝑋} ∩ ran 𝐹))
97, 8syl 17 . . . 4 (𝜑 → (𝐹 “ (𝐹 “ {𝑋})) = ({𝑋} ∩ ran 𝐹))
102snssd 4814 . . . . 5 (𝜑 → {𝑋} ⊆ ran 𝐹)
11 dfss2 3981 . . . . 5 ({𝑋} ⊆ ran 𝐹 ↔ ({𝑋} ∩ ran 𝐹) = {𝑋})
1210, 11sylib 218 . . . 4 (𝜑 → ({𝑋} ∩ ran 𝐹) = {𝑋})
139, 12eqtrd 2775 . . 3 (𝜑 → (𝐹 “ (𝐹 “ {𝑋})) = {𝑋})
14 funimacnv 6649 . . . . 5 (Fun 𝐹 → (𝐹 “ (𝐹 “ {𝑌})) = ({𝑌} ∩ ran 𝐹))
157, 14syl 17 . . . 4 (𝜑 → (𝐹 “ (𝐹 “ {𝑌})) = ({𝑌} ∩ ran 𝐹))
16 preimane.1 . . . . . 6 (𝜑𝑌 ∈ ran 𝐹)
1716snssd 4814 . . . . 5 (𝜑 → {𝑌} ⊆ ran 𝐹)
18 dfss2 3981 . . . . 5 ({𝑌} ⊆ ran 𝐹 ↔ ({𝑌} ∩ ran 𝐹) = {𝑌})
1917, 18sylib 218 . . . 4 (𝜑 → ({𝑌} ∩ ran 𝐹) = {𝑌})
2015, 19eqtrd 2775 . . 3 (𝜑 → (𝐹 “ (𝐹 “ {𝑌})) = {𝑌})
216, 13, 203netr4d 3016 . 2 (𝜑 → (𝐹 “ (𝐹 “ {𝑋})) ≠ (𝐹 “ (𝐹 “ {𝑌})))
22 imaeq2 6076 . . 3 ((𝐹 “ {𝑋}) = (𝐹 “ {𝑌}) → (𝐹 “ (𝐹 “ {𝑋})) = (𝐹 “ (𝐹 “ {𝑌})))
2322necon3i 2971 . 2 ((𝐹 “ (𝐹 “ {𝑋})) ≠ (𝐹 “ (𝐹 “ {𝑌})) → (𝐹 “ {𝑋}) ≠ (𝐹 “ {𝑌}))
2421, 23syl 17 1 (𝜑 → (𝐹 “ {𝑋}) ≠ (𝐹 “ {𝑌}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wne 2938  cin 3962  wss 3963  {csn 4631  ccnv 5688  ran crn 5690  cima 5692  Fun wfun 6557
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-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-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-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-fun 6565
This theorem is referenced by:  fnpreimac  32688
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