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Mirrors > Home > MPE Home > Th. List > fin23lem15 | Structured version Visualization version GIF version |
Description: Lemma for fin23 9799. 𝑈 is a monotone function. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
Ref | Expression |
---|---|
fin23lem.a | ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) |
Ref | Expression |
---|---|
fin23lem15 | ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (𝑈‘𝐴) ⊆ (𝑈‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6663 | . . 3 ⊢ (𝑏 = 𝐵 → (𝑈‘𝑏) = (𝑈‘𝐵)) | |
2 | 1 | sseq1d 3995 | . 2 ⊢ (𝑏 = 𝐵 → ((𝑈‘𝑏) ⊆ (𝑈‘𝐵) ↔ (𝑈‘𝐵) ⊆ (𝑈‘𝐵))) |
3 | fveq2 6663 | . . 3 ⊢ (𝑏 = 𝑎 → (𝑈‘𝑏) = (𝑈‘𝑎)) | |
4 | 3 | sseq1d 3995 | . 2 ⊢ (𝑏 = 𝑎 → ((𝑈‘𝑏) ⊆ (𝑈‘𝐵) ↔ (𝑈‘𝑎) ⊆ (𝑈‘𝐵))) |
5 | fveq2 6663 | . . 3 ⊢ (𝑏 = suc 𝑎 → (𝑈‘𝑏) = (𝑈‘suc 𝑎)) | |
6 | 5 | sseq1d 3995 | . 2 ⊢ (𝑏 = suc 𝑎 → ((𝑈‘𝑏) ⊆ (𝑈‘𝐵) ↔ (𝑈‘suc 𝑎) ⊆ (𝑈‘𝐵))) |
7 | fveq2 6663 | . . 3 ⊢ (𝑏 = 𝐴 → (𝑈‘𝑏) = (𝑈‘𝐴)) | |
8 | 7 | sseq1d 3995 | . 2 ⊢ (𝑏 = 𝐴 → ((𝑈‘𝑏) ⊆ (𝑈‘𝐵) ↔ (𝑈‘𝐴) ⊆ (𝑈‘𝐵))) |
9 | ssidd 3987 | . 2 ⊢ (𝐵 ∈ ω → (𝑈‘𝐵) ⊆ (𝑈‘𝐵)) | |
10 | fin23lem.a | . . . . 5 ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) | |
11 | 10 | fin23lem13 9742 | . . . 4 ⊢ (𝑎 ∈ ω → (𝑈‘suc 𝑎) ⊆ (𝑈‘𝑎)) |
12 | 11 | ad2antrr 722 | . . 3 ⊢ (((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) → (𝑈‘suc 𝑎) ⊆ (𝑈‘𝑎)) |
13 | sstr2 3971 | . . 3 ⊢ ((𝑈‘suc 𝑎) ⊆ (𝑈‘𝑎) → ((𝑈‘𝑎) ⊆ (𝑈‘𝐵) → (𝑈‘suc 𝑎) ⊆ (𝑈‘𝐵))) | |
14 | 12, 13 | syl 17 | . 2 ⊢ (((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) → ((𝑈‘𝑎) ⊆ (𝑈‘𝐵) → (𝑈‘suc 𝑎) ⊆ (𝑈‘𝐵))) |
15 | 2, 4, 6, 8, 9, 14 | findsg 7598 | 1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (𝑈‘𝐴) ⊆ (𝑈‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 ifcif 4463 ∪ cuni 4830 ran crn 5549 suc csuc 6186 ‘cfv 6348 ∈ cmpo 7147 ωcom 7569 seqωcseqom 8072 |
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-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 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-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-seqom 8073 |
This theorem is referenced by: fin23lem16 9745 |
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