![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fin23lem16 | Structured version Visualization version GIF version |
Description: Lemma for fin23 10428. 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 4946 | . . 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 8484 | . . . . 5 ⊢ 𝑈 Fn ω |
4 | fvelrnb 6962 | . . . . 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 7899 | . . . . . . . 8 ⊢ ∅ ∈ ω | |
7 | 0ss 4400 | . . . . . . . . 9 ⊢ ∅ ⊆ 𝑏 | |
8 | 2 | fin23lem15 10373 | . . . . . . . . 9 ⊢ (((𝑏 ∈ ω ∧ ∅ ∈ ω) ∧ ∅ ⊆ 𝑏) → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
9 | 7, 8 | mpan2 689 | . . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ ∅ ∈ ω) → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
10 | 6, 9 | mpan2 689 | . . . . . . 7 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
11 | vex 3465 | . . . . . . . . . 10 ⊢ 𝑡 ∈ V | |
12 | 11 | rnex 7922 | . . . . . . . . 9 ⊢ ran 𝑡 ∈ V |
13 | 12 | uniex 7751 | . . . . . . . 8 ⊢ ∪ ran 𝑡 ∈ V |
14 | 2 | seqom0g 8485 | . . . . . . . 8 ⊢ (∪ ran 𝑡 ∈ V → (𝑈‘∅) = ∪ ran 𝑡) |
15 | 13, 14 | ax-mp 5 | . . . . . . 7 ⊢ (𝑈‘∅) = ∪ ran 𝑡 |
16 | 10, 15 | sseqtrdi 4029 | . . . . . 6 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ ∪ ran 𝑡) |
17 | sseq1 4004 | . . . . . 6 ⊢ ((𝑈‘𝑏) = 𝑎 → ((𝑈‘𝑏) ⊆ ∪ ran 𝑡 ↔ 𝑎 ⊆ ∪ ran 𝑡)) | |
18 | 16, 17 | syl5ibcom 244 | . . . . 5 ⊢ (𝑏 ∈ ω → ((𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡)) |
19 | 18 | rexlimiv 3137 | . . . 4 ⊢ (∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡) |
20 | 5, 19 | sylbi 216 | . . 3 ⊢ (𝑎 ∈ ran 𝑈 → 𝑎 ⊆ ∪ ran 𝑡) |
21 | 1, 20 | mprgbir 3057 | . 2 ⊢ ∪ ran 𝑈 ⊆ ∪ ran 𝑡 |
22 | fnfvelrn 7093 | . . . . 5 ⊢ ((𝑈 Fn ω ∧ ∅ ∈ ω) → (𝑈‘∅) ∈ ran 𝑈) | |
23 | 3, 6, 22 | mp2an 690 | . . . 4 ⊢ (𝑈‘∅) ∈ ran 𝑈 |
24 | 15, 23 | eqeltrri 2822 | . . 3 ⊢ ∪ ran 𝑡 ∈ ran 𝑈 |
25 | elssuni 4944 | . . 3 ⊢ (∪ ran 𝑡 ∈ ran 𝑈 → ∪ ran 𝑡 ⊆ ∪ ran 𝑈) | |
26 | 24, 25 | ax-mp 5 | . 2 ⊢ ∪ ran 𝑡 ⊆ ∪ ran 𝑈 |
27 | 21, 26 | eqssi 3995 | 1 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 Vcvv 3461 ∩ cin 3945 ⊆ wss 3946 ∅c0 4324 ifcif 4532 ∪ cuni 4912 ran crn 5682 Fn wfn 6548 ‘cfv 6553 ∈ cmpo 7425 ωcom 7875 seqωcseqom 8476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-seqom 8477 |
This theorem is referenced by: fin23lem17 10377 fin23lem31 10382 |
Copyright terms: Public domain | W3C validator |