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 3991 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
2 | wnefimgd.2 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
3 | 2 | fdmd 6525 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
4 | 1, 3 | sseqtrrid 4022 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹) |
5 | sseqin2 4194 | . . . 4 ⊢ (𝐴 ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ 𝐴) = 𝐴) | |
6 | 4, 5 | sylib 220 | . . 3 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) = 𝐴) |
7 | wnefimgd.1 | . . 3 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
8 | 6, 7 | eqnetrd 3085 | . 2 ⊢ (𝜑 → (dom 𝐹 ∩ 𝐴) ≠ ∅) |
9 | 8 | imadisjlnd 40518 | 1 ⊢ (𝜑 → (𝐹 “ 𝐴) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ≠ wne 3018 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 dom cdm 5557 “ cima 5560 ⟶wf 6353 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-fn 6360 df-f 6361 |
This theorem is referenced by: imo72b2lem0 40523 imo72b2lem2 40525 imo72b2lem1 40528 imo72b2 40532 |
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