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Mirrors > Home > MPE Home > Th. List > mapdom3 | Structured version Visualization version GIF version |
Description: Set exponentiation dominates the base. (Contributed by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 17-Jul-2022.) |
Ref | Expression |
---|---|
mapdom3 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑m 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4342 | . . 3 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐵) | |
2 | simp1 1136 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑉) | |
3 | simp3 1138 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
4 | 2, 3 | mapsnend 9019 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝐴 ↑m {𝑥}) ≈ 𝐴) |
5 | 4 | ensymd 8984 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝐴 ≈ (𝐴 ↑m {𝑥})) |
6 | simp2 1137 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝐵 ∈ 𝑊) | |
7 | 3 | snssd 4805 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → {𝑥} ⊆ 𝐵) |
8 | ssdomg 8979 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝑊 → ({𝑥} ⊆ 𝐵 → {𝑥} ≼ 𝐵)) | |
9 | 6, 7, 8 | sylc 65 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → {𝑥} ≼ 𝐵) |
10 | vex 3477 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
11 | 10 | snnz 4773 | . . . . . . . 8 ⊢ {𝑥} ≠ ∅ |
12 | simpl 483 | . . . . . . . . 9 ⊢ (({𝑥} = ∅ ∧ 𝐴 = ∅) → {𝑥} = ∅) | |
13 | 12 | necon3ai 2964 | . . . . . . . 8 ⊢ ({𝑥} ≠ ∅ → ¬ ({𝑥} = ∅ ∧ 𝐴 = ∅)) |
14 | 11, 13 | ax-mp 5 | . . . . . . 7 ⊢ ¬ ({𝑥} = ∅ ∧ 𝐴 = ∅) |
15 | mapdom2 9131 | . . . . . . 7 ⊢ (({𝑥} ≼ 𝐵 ∧ ¬ ({𝑥} = ∅ ∧ 𝐴 = ∅)) → (𝐴 ↑m {𝑥}) ≼ (𝐴 ↑m 𝐵)) | |
16 | 9, 14, 15 | sylancl 586 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → (𝐴 ↑m {𝑥}) ≼ (𝐴 ↑m 𝐵)) |
17 | endomtr 8991 | . . . . . 6 ⊢ ((𝐴 ≈ (𝐴 ↑m {𝑥}) ∧ (𝐴 ↑m {𝑥}) ≼ (𝐴 ↑m 𝐵)) → 𝐴 ≼ (𝐴 ↑m 𝐵)) | |
18 | 5, 16, 17 | syl2anc 584 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑥 ∈ 𝐵) → 𝐴 ≼ (𝐴 ↑m 𝐵)) |
19 | 18 | 3expia 1121 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐵 → 𝐴 ≼ (𝐴 ↑m 𝐵))) |
20 | 19 | exlimdv 1936 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 ∈ 𝐵 → 𝐴 ≼ (𝐴 ↑m 𝐵))) |
21 | 1, 20 | biimtrid 241 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ≠ ∅ → 𝐴 ≼ (𝐴 ↑m 𝐵))) |
22 | 21 | 3impia 1117 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑m 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2939 ⊆ wss 3944 ∅c0 4318 {csn 4622 class class class wbr 5141 (class class class)co 7393 ↑m cmap 8803 ≈ cen 8919 ≼ cdom 8920 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 |
This theorem is referenced by: infmap2 10195 |
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