Theorem preiman0 31021

Description: The preimage of a nonempty set is nonempty. (Contributed by Thierry Arnoux, 9-Jun-2024.)

Assertion

Ref	Expression
preiman0	⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ 𝐴) ≠ ∅)

Proof of Theorem preiman0

Step	Hyp	Ref	Expression
1		df-rn 5599	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
2	1	ineq1i 4147	. . . . 5 ⊢ (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (dom ◡𝐹 ∩ (𝐴 ∩ ran 𝐹))
3		df-ss 3908	. . . . . . . . 9 ⊢ (𝐴 ⊆ ran 𝐹 ↔ (𝐴 ∩ ran 𝐹) = 𝐴)
4	3	biimpi 215	. . . . . . . 8 ⊢ (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
5	4	ineq2d 4151	. . . . . . 7 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (ran 𝐹 ∩ 𝐴))
6		sseqin2 4154	. . . . . . . 8 ⊢ (𝐴 ⊆ ran 𝐹 ↔ (ran 𝐹 ∩ 𝐴) = 𝐴)
7	6	biimpi 215	. . . . . . 7 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ 𝐴) = 𝐴)
8	5, 7	eqtrd 2779	. . . . . 6 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = 𝐴)
9	8	3ad2ant2 1132	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = 𝐴)
10		fimacnvinrn 6943	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))
11	10	eqeq1d 2741	. . . . . . . 8 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) = ∅ ↔ (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅))
12	11	biimpa 476	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)
13	12	3adant2 1129	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)
14		imadisj 5985	. . . . . 6 ⊢ ((◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅ ↔ (dom ◡𝐹 ∩ (𝐴 ∩ ran 𝐹)) = ∅)
15	13, 14	sylib 217	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → (dom ◡𝐹 ∩ (𝐴 ∩ ran 𝐹)) = ∅)
16	2, 9, 15	3eqtr3a 2803	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → 𝐴 = ∅)
17	16	3expia 1119	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → ((◡𝐹 “ 𝐴) = ∅ → 𝐴 = ∅))
18	17	necon3d 2965	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → (𝐴 ≠ ∅ → (◡𝐹 “ 𝐴) ≠ ∅))
19	18	3impia 1115	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ 𝐴) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1541   ≠ wne 2944   ∩ cin 3890   ⊆ wss 3891  ∅c0 4261  ^◡ccnv 5587  dom cdm 5588  ran crn 5589   “ cima 5591  Fun wfun 6424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-fun 6432  df-fn 6433  df-f 6434  df-fo 6436

This theorem is referenced by:  zarcmplem  31810