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Theorem preiman0 31329
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 5631 . . . . . 6 ran 𝐹 = dom 𝐹
21ineq1i 4155 . . . . 5 (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (dom 𝐹 ∩ (𝐴 ∩ ran 𝐹))
3 df-ss 3915 . . . . . . . . 9 (𝐴 ⊆ ran 𝐹 ↔ (𝐴 ∩ ran 𝐹) = 𝐴)
43biimpi 215 . . . . . . . 8 (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
54ineq2d 4159 . . . . . . 7 (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (ran 𝐹𝐴))
6 sseqin2 4162 . . . . . . . 8 (𝐴 ⊆ ran 𝐹 ↔ (ran 𝐹𝐴) = 𝐴)
76biimpi 215 . . . . . . 7 (𝐴 ⊆ ran 𝐹 → (ran 𝐹𝐴) = 𝐴)
85, 7eqtrd 2776 . . . . . 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 7005 . . . . . . . . 9 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
1110eqeq1d 2738 . . . . . . . 8 (Fun 𝐹 → ((𝐹𝐴) = ∅ ↔ (𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅))
1211biimpa 477 . . . . . . 7 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)
13123adant2 1130 . . . . . 6 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)
14 imadisj 6018 . . . . . 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 2800 . . . 4 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → 𝐴 = ∅)
17163expia 1120 . . 3 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) = ∅ → 𝐴 = ∅))
1817necon3d 2961 . 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 396  w3a 1086   = wceq 1540  wne 2940  cin 3897  wss 3898  c0 4269  ccnv 5619  dom cdm 5620  ran crn 5621  cima 5623  Fun wfun 6473
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 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-fun 6481  df-fn 6482  df-f 6483  df-fo 6485
This theorem is referenced by:  zarcmplem  32129
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