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Mirrors > Home > MPE Home > Th. List > smoiun | Structured version Visualization version GIF version |
Description: The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011.) |
Ref | Expression |
---|---|
smoiun | ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) ⊆ (𝐵‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliun 4885 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵‘𝑥)) | |
2 | smofvon 7979 | . . . . 5 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝐵‘𝐴) ∈ On) | |
3 | smoel 7980 | . . . . . 6 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐵‘𝑥) ∈ (𝐵‘𝐴)) | |
4 | 3 | 3expia 1118 | . . . . 5 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝑥 ∈ 𝐴 → (𝐵‘𝑥) ∈ (𝐵‘𝐴))) |
5 | ontr1 6205 | . . . . . 6 ⊢ ((𝐵‘𝐴) ∈ On → ((𝑦 ∈ (𝐵‘𝑥) ∧ (𝐵‘𝑥) ∈ (𝐵‘𝐴)) → 𝑦 ∈ (𝐵‘𝐴))) | |
6 | 5 | expcomd 420 | . . . . 5 ⊢ ((𝐵‘𝐴) ∈ On → ((𝐵‘𝑥) ∈ (𝐵‘𝐴) → (𝑦 ∈ (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴)))) |
7 | 2, 4, 6 | sylsyld 61 | . . . 4 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝑥 ∈ 𝐴 → (𝑦 ∈ (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴)))) |
8 | 7 | rexlimdv 3242 | . . 3 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴))) |
9 | 1, 8 | syl5bi 245 | . 2 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) → 𝑦 ∈ (𝐵‘𝐴))) |
10 | 9 | ssrdv 3921 | 1 ⊢ ((Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵) → ∪ 𝑥 ∈ 𝐴 (𝐵‘𝑥) ⊆ (𝐵‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∃wrex 3107 ⊆ wss 3881 ∪ ciun 4881 dom cdm 5519 Oncon0 6159 ‘cfv 6324 Smo wsmo 7965 |
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-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 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-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-ord 6162 df-on 6163 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-smo 7966 |
This theorem is referenced by: (None) |
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