Theorem preiman0 33036

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 5674	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
2	1	ineq1i 4170	. . . . 5 ⊢ (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (dom ◡𝐹 ∩ (𝐴 ∩ ran 𝐹))
3		dfss2 3924	. . . . . . . . 9 ⊢ (𝐴 ⊆ ran 𝐹 ↔ (𝐴 ∩ ran 𝐹) = 𝐴)
4	3	biimpi 219	. . . . . . . 8 ⊢ (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
5	4	ineq2d 4174	. . . . . . 7 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (ran 𝐹 ∩ 𝐴))
6		sseqin2 4177	. . . . . . . 8 ⊢ (𝐴 ⊆ ran 𝐹 ↔ (ran 𝐹 ∩ 𝐴) = 𝐴)
7	6	biimpi 219	. . . . . . 7 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ 𝐴) = 𝐴)
8	5, 7	eqtrd 2798	. . . . . 6 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = 𝐴)
9	8	3ad2ant2 1152	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = 𝐴)
10		fimacnvinrn 7068	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))
11	10	eqeq1d 2765	. . . . . . . 8 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) = ∅ ↔ (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅))
12	11	biimpa 481	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)
13	12	3adant2 1149	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)
14		imadisj 6084	. . . . . 6 ⊢ ((◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅ ↔ (dom ◡𝐹 ∩ (𝐴 ∩ ran 𝐹)) = ∅)
15	13, 14	sylib 221	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → (dom ◡𝐹 ∩ (𝐴 ∩ ran 𝐹)) = ∅)
16	2, 9, 15	3eqtr3a 2822	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → 𝐴 = ∅)
17	16	3expia 1139	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → ((◡𝐹 “ 𝐴) = ∅ → 𝐴 = ∅))
18	17	necon3d 2979	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → (𝐴 ≠ ∅ → (◡𝐹 “ 𝐴) ≠ ∅))
19	18	3impia 1135	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ 𝐴) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ≠ wne 2958   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   “ cima 5666  Fun wfun 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544

This theorem is referenced by:  zarcmplem  34252