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Theorem preimane 30531
 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 4727 . . . . . 6 (𝑋 ∈ ran 𝐹 → ({𝑋} = {𝑌} → 𝑋 = 𝑌))
42, 3syl 17 . . . . 5 (𝜑 → ({𝑋} = {𝑌} → 𝑋 = 𝑌))
54necon3d 2972 . . . 4 (𝜑 → (𝑋𝑌 → {𝑋} ≠ {𝑌}))
61, 5mpd 15 . . 3 (𝜑 → {𝑋} ≠ {𝑌})
7 preimane.f . . . . 5 (𝜑 → Fun 𝐹)
8 funimacnv 6416 . . . . 5 (Fun 𝐹 → (𝐹 “ (𝐹 “ {𝑋})) = ({𝑋} ∩ ran 𝐹))
97, 8syl 17 . . . 4 (𝜑 → (𝐹 “ (𝐹 “ {𝑋})) = ({𝑋} ∩ ran 𝐹))
102snssd 4699 . . . . 5 (𝜑 → {𝑋} ⊆ ran 𝐹)
11 df-ss 3875 . . . . 5 ({𝑋} ⊆ ran 𝐹 ↔ ({𝑋} ∩ ran 𝐹) = {𝑋})
1210, 11sylib 221 . . . 4 (𝜑 → ({𝑋} ∩ ran 𝐹) = {𝑋})
139, 12eqtrd 2793 . . 3 (𝜑 → (𝐹 “ (𝐹 “ {𝑋})) = {𝑋})
14 funimacnv 6416 . . . . 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 4699 . . . . 5 (𝜑 → {𝑌} ⊆ ran 𝐹)
18 df-ss 3875 . . . . 5 ({𝑌} ⊆ ran 𝐹 ↔ ({𝑌} ∩ ran 𝐹) = {𝑌})
1917, 18sylib 221 . . . 4 (𝜑 → ({𝑌} ∩ ran 𝐹) = {𝑌})
2015, 19eqtrd 2793 . . 3 (𝜑 → (𝐹 “ (𝐹 “ {𝑌})) = {𝑌})
216, 13, 203netr4d 3028 . 2 (𝜑 → (𝐹 “ (𝐹 “ {𝑋})) ≠ (𝐹 “ (𝐹 “ {𝑌})))
22 imaeq2 5897 . . 3 ((𝐹 “ {𝑋}) = (𝐹 “ {𝑌}) → (𝐹 “ (𝐹 “ {𝑋})) = (𝐹 “ (𝐹 “ {𝑌})))
2322necon3i 2983 . 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 1538   ∈ wcel 2111   ≠ wne 2951   ∩ cin 3857   ⊆ wss 3858  {csn 4522  ◡ccnv 5523  ran crn 5525   “ cima 5527  Fun wfun 6329 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 5169  ax-nul 5176  ax-pr 5298 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-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-fun 6337 This theorem is referenced by:  fnpreimac  30532
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