![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fin23lem16 | Structured version Visualization version GIF version |
Description: Lemma for fin23 9499. 𝑈 ranges over the original set; in particular ran 𝑈 is a set, although we do not assume here that 𝑈 is. (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 |
---|---|
fin23lem16 | ⊢ ∪ ran 𝑈 = ∪ ran 𝑡 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unissb 4661 | . . 3 ⊢ (∪ ran 𝑈 ⊆ ∪ ran 𝑡 ↔ ∀𝑎 ∈ ran 𝑈 𝑎 ⊆ ∪ ran 𝑡) | |
2 | fin23lem.a | . . . . . 6 ⊢ 𝑈 = seq𝜔((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) | |
3 | 2 | fnseqom 7789 | . . . . 5 ⊢ 𝑈 Fn ω |
4 | fvelrnb 6468 | . . . . 5 ⊢ (𝑈 Fn ω → (𝑎 ∈ ran 𝑈 ↔ ∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎)) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (𝑎 ∈ ran 𝑈 ↔ ∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎) |
6 | peano1 7319 | . . . . . . . 8 ⊢ ∅ ∈ ω | |
7 | 0ss 4168 | . . . . . . . . 9 ⊢ ∅ ⊆ 𝑏 | |
8 | 2 | fin23lem15 9444 | . . . . . . . . 9 ⊢ (((𝑏 ∈ ω ∧ ∅ ∈ ω) ∧ ∅ ⊆ 𝑏) → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
9 | 7, 8 | mpan2 683 | . . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ ∅ ∈ ω) → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
10 | 6, 9 | mpan2 683 | . . . . . . 7 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
11 | vex 3388 | . . . . . . . . . 10 ⊢ 𝑡 ∈ V | |
12 | 11 | rnex 7335 | . . . . . . . . 9 ⊢ ran 𝑡 ∈ V |
13 | 12 | uniex 7187 | . . . . . . . 8 ⊢ ∪ ran 𝑡 ∈ V |
14 | 2 | seqom0g 7790 | . . . . . . . 8 ⊢ (∪ ran 𝑡 ∈ V → (𝑈‘∅) = ∪ ran 𝑡) |
15 | 13, 14 | ax-mp 5 | . . . . . . 7 ⊢ (𝑈‘∅) = ∪ ran 𝑡 |
16 | 10, 15 | syl6sseq 3847 | . . . . . 6 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ ∪ ran 𝑡) |
17 | sseq1 3822 | . . . . . 6 ⊢ ((𝑈‘𝑏) = 𝑎 → ((𝑈‘𝑏) ⊆ ∪ ran 𝑡 ↔ 𝑎 ⊆ ∪ ran 𝑡)) | |
18 | 16, 17 | syl5ibcom 237 | . . . . 5 ⊢ (𝑏 ∈ ω → ((𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡)) |
19 | 18 | rexlimiv 3208 | . . . 4 ⊢ (∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡) |
20 | 5, 19 | sylbi 209 | . . 3 ⊢ (𝑎 ∈ ran 𝑈 → 𝑎 ⊆ ∪ ran 𝑡) |
21 | 1, 20 | mprgbir 3108 | . 2 ⊢ ∪ ran 𝑈 ⊆ ∪ ran 𝑡 |
22 | fnfvelrn 6582 | . . . . 5 ⊢ ((𝑈 Fn ω ∧ ∅ ∈ ω) → (𝑈‘∅) ∈ ran 𝑈) | |
23 | 3, 6, 22 | mp2an 684 | . . . 4 ⊢ (𝑈‘∅) ∈ ran 𝑈 |
24 | 15, 23 | eqeltrri 2875 | . . 3 ⊢ ∪ ran 𝑡 ∈ ran 𝑈 |
25 | elssuni 4659 | . . 3 ⊢ (∪ ran 𝑡 ∈ ran 𝑈 → ∪ ran 𝑡 ⊆ ∪ ran 𝑈) | |
26 | 24, 25 | ax-mp 5 | . 2 ⊢ ∪ ran 𝑡 ⊆ ∪ ran 𝑈 |
27 | 21, 26 | eqssi 3814 | 1 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∃wrex 3090 Vcvv 3385 ∩ cin 3768 ⊆ wss 3769 ∅c0 4115 ifcif 4277 ∪ cuni 4628 ran crn 5313 Fn wfn 6096 ‘cfv 6101 ↦ cmpt2 6880 ωcom 7299 seq𝜔cseqom 7781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-seqom 7782 |
This theorem is referenced by: fin23lem17 9448 fin23lem31 9453 |
Copyright terms: Public domain | W3C validator |