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| Mirrors > Home > MPE Home > Th. List > Mathboxes > wnefimgd | Structured version Visualization version GIF version | ||
| Description: The image of a mapping from A is nonempty if A is nonempty. (Contributed by Stanislas Polu, 9-Mar-2020.) | 
| Ref | Expression | 
|---|---|
| wnefimgd.1 | ⊢ (𝜑 → 𝐴 ≠ ∅) | 
| wnefimgd.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | 
| Ref | Expression | 
|---|---|
| wnefimgd | ⊢ (𝜑 → (𝐹 “ 𝐴) ≠ ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssid 4005 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
| 2 | wnefimgd.2 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
| 3 | 2 | fdmd 6745 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) | 
| 4 | 1, 3 | sseqtrrid 4026 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) | 
| 5 | sseqin2 4222 | . . . 4 ⊢ (𝐴 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝐴) = 𝐴) | |
| 6 | 4, 5 | sylib 218 | . . 3 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) = 𝐴) | 
| 7 | wnefimgd.1 | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
| 8 | 6, 7 | eqnetrd 3007 | . 2 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) ≠ ∅) | 
| 9 | 8 | imadisjlnd 6098 | 1 ⊢ (𝜑 → (𝐹 “ 𝐴) ≠ ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ≠ wne 2939 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 dom cdm 5684 “ cima 5687 ⟶wf 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: imo72b2lem0 44183 imo72b2lem2 44185 imo72b2lem1 44187 imo72b2 44190 | 
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