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Theorem preiman0 32719
Description: The preimage of a nonempty set is nonempty. (Contributed by Thierry Arnoux, 9-Jun-2024.)
Assertion
Ref Expression
preiman0 ((Fun 𝐹𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → (𝐹𝐴) ≠ ∅)

Proof of Theorem preiman0
StepHypRef Expression
1 df-rn 5696 . . . . . 6 ran 𝐹 = dom 𝐹
21ineq1i 4216 . . . . 5 (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (dom 𝐹 ∩ (𝐴 ∩ ran 𝐹))
3 dfss2 3969 . . . . . . . . 9 (𝐴 ⊆ ran 𝐹 ↔ (𝐴 ∩ ran 𝐹) = 𝐴)
43biimpi 216 . . . . . . . 8 (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
54ineq2d 4220 . . . . . . 7 (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (ran 𝐹𝐴))
6 sseqin2 4223 . . . . . . . 8 (𝐴 ⊆ ran 𝐹 ↔ (ran 𝐹𝐴) = 𝐴)
76biimpi 216 . . . . . . 7 (𝐴 ⊆ ran 𝐹 → (ran 𝐹𝐴) = 𝐴)
85, 7eqtrd 2777 . . . . . 6 (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = 𝐴)
983ad2ant2 1135 . . . . 5 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = 𝐴)
10 fimacnvinrn 7091 . . . . . . . . 9 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
1110eqeq1d 2739 . . . . . . . 8 (Fun 𝐹 → ((𝐹𝐴) = ∅ ↔ (𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅))
1211biimpa 476 . . . . . . 7 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)
13123adant2 1132 . . . . . 6 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)
14 imadisj 6098 . . . . . 6 ((𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅ ↔ (dom 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = ∅)
1513, 14sylib 218 . . . . 5 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → (dom 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = ∅)
162, 9, 153eqtr3a 2801 . . . 4 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → 𝐴 = ∅)
17163expia 1122 . . 3 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) = ∅ → 𝐴 = ∅))
1817necon3d 2961 . 2 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → (𝐴 ≠ ∅ → (𝐹𝐴) ≠ ∅))
19183impia 1118 1 ((Fun 𝐹𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → (𝐹𝐴) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wne 2940  cin 3950  wss 3951  c0 4333  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688  Fun wfun 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567
This theorem is referenced by:  zarcmplem  33880
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