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Theorem preiman0 32199
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 5687 . . . . . 6 ran 𝐹 = dom 𝐹
21ineq1i 4208 . . . . 5 (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (dom 𝐹 ∩ (𝐴 ∩ ran 𝐹))
3 df-ss 3965 . . . . . . . . 9 (𝐴 ⊆ ran 𝐹 ↔ (𝐴 ∩ ran 𝐹) = 𝐴)
43biimpi 215 . . . . . . . 8 (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
54ineq2d 4212 . . . . . . 7 (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (ran 𝐹𝐴))
6 sseqin2 4215 . . . . . . . 8 (𝐴 ⊆ ran 𝐹 ↔ (ran 𝐹𝐴) = 𝐴)
76biimpi 215 . . . . . . 7 (𝐴 ⊆ ran 𝐹 → (ran 𝐹𝐴) = 𝐴)
85, 7eqtrd 2771 . . . . . 6 (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = 𝐴)
983ad2ant2 1133 . . . . 5 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = 𝐴)
10 fimacnvinrn 7073 . . . . . . . . 9 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
1110eqeq1d 2733 . . . . . . . 8 (Fun 𝐹 → ((𝐹𝐴) = ∅ ↔ (𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅))
1211biimpa 476 . . . . . . 7 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)
13123adant2 1130 . . . . . 6 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)
14 imadisj 6079 . . . . . 6 ((𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅ ↔ (dom 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = ∅)
1513, 14sylib 217 . . . . 5 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → (dom 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = ∅)
162, 9, 153eqtr3a 2795 . . . 4 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → 𝐴 = ∅)
17163expia 1120 . . 3 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) = ∅ → 𝐴 = ∅))
1817necon3d 2960 . 2 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → (𝐴 ≠ ∅ → (𝐹𝐴) ≠ ∅))
19183impia 1116 1 ((Fun 𝐹𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → (𝐹𝐴) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wne 2939  cin 3947  wss 3948  c0 4322  ccnv 5675  dom cdm 5676  ran crn 5677  cima 5679  Fun wfun 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549
This theorem is referenced by:  zarcmplem  33160
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