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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 3937 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
2 | wnefimgd.2 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
3 | 2 | fdmd 6497 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
4 | 1, 3 | sseqtrrid 3968 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) |
5 | sseqin2 4142 | . . . 4 ⊢ (𝐴 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝐴) = 𝐴) | |
6 | 4, 5 | sylib 221 | . . 3 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) = 𝐴) |
7 | wnefimgd.1 | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
8 | 6, 7 | eqnetrd 3054 | . 2 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) ≠ ∅) |
9 | 8 | imadisjlnd 40864 | 1 ⊢ (𝜑 → (𝐹 “ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ≠ wne 2987 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 dom cdm 5519 “ cima 5522 ⟶wf 6320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fn 6327 df-f 6328 |
This theorem is referenced by: imo72b2lem0 40869 imo72b2lem2 40871 imo72b2lem1 40874 imo72b2 40878 |
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