Theorem preiman0 32725

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 5700	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
2	1	ineq1i 4224	. . . . 5 ⊢ (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (dom ◡𝐹 ∩ (𝐴 ∩ ran 𝐹))
3		dfss2 3981	. . . . . . . . 9 ⊢ (𝐴 ⊆ ran 𝐹 ↔ (𝐴 ∩ ran 𝐹) = 𝐴)
4	3	biimpi 216	. . . . . . . 8 ⊢ (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
5	4	ineq2d 4228	. . . . . . 7 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (ran 𝐹 ∩ 𝐴))
6		sseqin2 4231	. . . . . . . 8 ⊢ (𝐴 ⊆ ran 𝐹 ↔ (ran 𝐹 ∩ 𝐴) = 𝐴)
7	6	biimpi 216	. . . . . . 7 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ 𝐴) = 𝐴)
8	5, 7	eqtrd 2775	. . . . . 6 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = 𝐴)
9	8	3ad2ant2 1133	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = 𝐴)
10		fimacnvinrn 7091	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)))
11	10	eqeq1d 2737	. . . . . . . 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 1130	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)
14		imadisj 6100	. . . . . 6 ⊢ ((◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅ ↔ (dom ◡𝐹 ∩ (𝐴 ∩ ran 𝐹)) = ∅)
15	13, 14	sylib 218	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → (dom ◡𝐹 ∩ (𝐴 ∩ ran 𝐹)) = ∅)
16	2, 9, 15	3eqtr3a 2799	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ (◡𝐹 “ 𝐴) = ∅) → 𝐴 = ∅)
17	16	3expia 1120	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → ((◡𝐹 “ 𝐴) = ∅ → 𝐴 = ∅))
18	17	necon3d 2959	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → (𝐴 ≠ ∅ → (◡𝐹 “ 𝐴) ≠ ∅))
19	18	3impia 1116	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ 𝐴) ≠ ∅)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ≠ wne 2938   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  ^◡ccnv 5688  dom cdm 5689  ran crn 5690   “ cima 5692  Fun wfun 6557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569

This theorem is referenced by:  zarcmplem  33842