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Theorem preiman0 30472
 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 5534 . . . . . 6 ran 𝐹 = dom 𝐹
21ineq1i 4138 . . . . 5 (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (dom 𝐹 ∩ (𝐴 ∩ ran 𝐹))
3 df-ss 3901 . . . . . . . . 9 (𝐴 ⊆ ran 𝐹 ↔ (𝐴 ∩ ran 𝐹) = 𝐴)
43biimpi 219 . . . . . . . 8 (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
54ineq2d 4142 . . . . . . 7 (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (ran 𝐹𝐴))
6 sseqin2 4145 . . . . . . . 8 (𝐴 ⊆ ran 𝐹 ↔ (ran 𝐹𝐴) = 𝐴)
76biimpi 219 . . . . . . 7 (𝐴 ⊆ ran 𝐹 → (ran 𝐹𝐴) = 𝐴)
85, 7eqtrd 2836 . . . . . 6 (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = 𝐴)
983ad2ant2 1131 . . . . 5 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = 𝐴)
10 fimacnvinrn 6821 . . . . . . . . 9 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
1110eqeq1d 2803 . . . . . . . 8 (Fun 𝐹 → ((𝐹𝐴) = ∅ ↔ (𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅))
1211biimpa 480 . . . . . . 7 ((Fun 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)
13123adant2 1128 . . . . . 6 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → (𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)
14 imadisj 5919 . . . . . 6 ((𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅ ↔ (dom 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = ∅)
1513, 14sylib 221 . . . . 5 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → (dom 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = ∅)
162, 9, 153eqtr3a 2860 . . . 4 ((Fun 𝐹𝐴 ⊆ ran 𝐹 ∧ (𝐹𝐴) = ∅) → 𝐴 = ∅)
17163expia 1118 . . 3 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) = ∅ → 𝐴 = ∅))
1817necon3d 3011 . 2 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → (𝐴 ≠ ∅ → (𝐹𝐴) ≠ ∅))
19183impia 1114 1 ((Fun 𝐹𝐴 ⊆ ran 𝐹𝐴 ≠ ∅) → (𝐹𝐴) ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ≠ wne 2990   ∩ cin 3883   ⊆ wss 3884  ∅c0 4246  ◡ccnv 5522  dom cdm 5523  ran crn 5524   “ cima 5526  Fun wfun 6322 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fo 6334  df-fv 6336 This theorem is referenced by:  zarcmplem  31234
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