Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > preiman0 | Structured version Visualization version GIF version |
Description: The preimage of a nonempty set is nonempty. (Contributed by Thierry Arnoux, 9-Jun-2024.) |
Ref | Expression |
---|---|
preiman0 | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ 𝐴) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rn 5631 | . . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹 | |
2 | 1 | ineq1i 4155 | . . . . 5 ⊢ (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (dom ◡𝐹 ∩ (𝐴 ∩ ran 𝐹)) |
3 | df-ss 3915 | . . . . . . . . 9 ⊢ (𝐴 ⊆ ran 𝐹 ↔ (𝐴 ∩ ran 𝐹) = 𝐴) | |
4 | 3 | biimpi 215 | . . . . . . . 8 ⊢ (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴) |
5 | 4 | ineq2d 4159 | . . . . . . 7 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ (𝐴 ∩ ran 𝐹)) = (ran 𝐹 ∩ 𝐴)) |
6 | sseqin2 4162 | . . . . . . . 8 ⊢ (𝐴 ⊆ ran 𝐹 ↔ (ran 𝐹 ∩ 𝐴) = 𝐴) | |
7 | 6 | biimpi 215 | . . . . . . 7 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ 𝐴) = 𝐴) |
8 | 5, 7 | eqtrd 2776 | . . . . . 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 7005 | . . . . . . . . 9 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) | |
11 | 10 | eqeq1d 2738 | . . . . . . . 8 ⊢ (Fun 𝐹 → ((◡𝐹 “ 𝐴) = ∅ ↔ (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∅)) |
12 | 11 | biimpa 477 | . . . . . . 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 6018 | . . . . . 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 2800 | . . . 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 2961 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → (𝐴 ≠ ∅ → (◡𝐹 “ 𝐴) ≠ ∅)) |
19 | 18 | 3impia 1116 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ 𝐴) ≠ ∅) |
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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