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Mirrors > Home > MPE Home > Th. List > fin23lem15 | Structured version Visualization version GIF version |
Description: Lemma for fin23 9811. 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 6670 | . . 3 ⊢ (𝑏 = 𝐵 → (𝑈‘𝑏) = (𝑈‘𝐵)) | |
2 | 1 | sseq1d 3998 | . 2 ⊢ (𝑏 = 𝐵 → ((𝑈‘𝑏) ⊆ (𝑈‘𝐵) ↔ (𝑈‘𝐵) ⊆ (𝑈‘𝐵))) |
3 | fveq2 6670 | . . 3 ⊢ (𝑏 = 𝑎 → (𝑈‘𝑏) = (𝑈‘𝑎)) | |
4 | 3 | sseq1d 3998 | . 2 ⊢ (𝑏 = 𝑎 → ((𝑈‘𝑏) ⊆ (𝑈‘𝐵) ↔ (𝑈‘𝑎) ⊆ (𝑈‘𝐵))) |
5 | fveq2 6670 | . . 3 ⊢ (𝑏 = suc 𝑎 → (𝑈‘𝑏) = (𝑈‘suc 𝑎)) | |
6 | 5 | sseq1d 3998 | . 2 ⊢ (𝑏 = suc 𝑎 → ((𝑈‘𝑏) ⊆ (𝑈‘𝐵) ↔ (𝑈‘suc 𝑎) ⊆ (𝑈‘𝐵))) |
7 | fveq2 6670 | . . 3 ⊢ (𝑏 = 𝐴 → (𝑈‘𝑏) = (𝑈‘𝐴)) | |
8 | 7 | sseq1d 3998 | . 2 ⊢ (𝑏 = 𝐴 → ((𝑈‘𝑏) ⊆ (𝑈‘𝐵) ↔ (𝑈‘𝐴) ⊆ (𝑈‘𝐵))) |
9 | ssidd 3990 | . 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 9754 | . . . 4 ⊢ (𝑎 ∈ ω → (𝑈‘suc 𝑎) ⊆ (𝑈‘𝑎)) |
12 | 11 | ad2antrr 724 | . . 3 ⊢ (((𝑎 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑎) → (𝑈‘suc 𝑎) ⊆ (𝑈‘𝑎)) |
13 | sstr2 3974 | . . 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 7609 | 1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (𝑈‘𝐴) ⊆ (𝑈‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 ifcif 4467 ∪ cuni 4838 ran crn 5556 suc csuc 6193 ‘cfv 6355 ∈ cmpo 7158 ωcom 7580 seqωcseqom 8083 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-seqom 8084 |
This theorem is referenced by: fin23lem16 9757 |
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