Mathbox for Stanislas Polu |
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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 3986 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
2 | wnefimgd.2 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
3 | 2 | fdmd 6516 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
4 | 1, 3 | sseqtrrid 4017 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) |
5 | sseqin2 4189 | . . . 4 ⊢ (𝐴 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝐴) = 𝐴) | |
6 | 4, 5 | sylib 219 | . . 3 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) = 𝐴) |
7 | wnefimgd.1 | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
8 | 6, 7 | eqnetrd 3080 | . 2 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) ≠ ∅) |
9 | 8 | imadisjlnd 40389 | 1 ⊢ (𝜑 → (𝐹 “ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ≠ wne 3013 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 dom cdm 5548 “ cima 5551 ⟶wf 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-fn 6351 df-f 6352 |
This theorem is referenced by: imo72b2lem0 40394 imo72b2lem2 40396 imo72b2lem1 40399 imo72b2 40403 |
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