![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fin23lem16 | Structured version Visualization version GIF version |
Description: Lemma for fin23 9800. 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 4832 | . . 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 8074 | . . . . 5 ⊢ 𝑈 Fn ω |
4 | fvelrnb 6701 | . . . . 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 7581 | . . . . . . . 8 ⊢ ∅ ∈ ω | |
7 | 0ss 4304 | . . . . . . . . 9 ⊢ ∅ ⊆ 𝑏 | |
8 | 2 | fin23lem15 9745 | . . . . . . . . 9 ⊢ (((𝑏 ∈ ω ∧ ∅ ∈ ω) ∧ ∅ ⊆ 𝑏) → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
9 | 7, 8 | mpan2 690 | . . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ ∅ ∈ ω) → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
10 | 6, 9 | mpan2 690 | . . . . . . 7 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
11 | vex 3444 | . . . . . . . . . 10 ⊢ 𝑡 ∈ V | |
12 | 11 | rnex 7599 | . . . . . . . . 9 ⊢ ran 𝑡 ∈ V |
13 | 12 | uniex 7447 | . . . . . . . 8 ⊢ ∪ ran 𝑡 ∈ V |
14 | 2 | seqom0g 8075 | . . . . . . . 8 ⊢ (∪ ran 𝑡 ∈ V → (𝑈‘∅) = ∪ ran 𝑡) |
15 | 13, 14 | ax-mp 5 | . . . . . . 7 ⊢ (𝑈‘∅) = ∪ ran 𝑡 |
16 | 10, 15 | sseqtrdi 3965 | . . . . . 6 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ ∪ ran 𝑡) |
17 | sseq1 3940 | . . . . . 6 ⊢ ((𝑈‘𝑏) = 𝑎 → ((𝑈‘𝑏) ⊆ ∪ ran 𝑡 ↔ 𝑎 ⊆ ∪ ran 𝑡)) | |
18 | 16, 17 | syl5ibcom 248 | . . . . 5 ⊢ (𝑏 ∈ ω → ((𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡)) |
19 | 18 | rexlimiv 3239 | . . . 4 ⊢ (∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡) |
20 | 5, 19 | sylbi 220 | . . 3 ⊢ (𝑎 ∈ ran 𝑈 → 𝑎 ⊆ ∪ ran 𝑡) |
21 | 1, 20 | mprgbir 3121 | . 2 ⊢ ∪ ran 𝑈 ⊆ ∪ ran 𝑡 |
22 | fnfvelrn 6825 | . . . . 5 ⊢ ((𝑈 Fn ω ∧ ∅ ∈ ω) → (𝑈‘∅) ∈ ran 𝑈) | |
23 | 3, 6, 22 | mp2an 691 | . . . 4 ⊢ (𝑈‘∅) ∈ ran 𝑈 |
24 | 15, 23 | eqeltrri 2887 | . . 3 ⊢ ∪ ran 𝑡 ∈ ran 𝑈 |
25 | elssuni 4830 | . . 3 ⊢ (∪ ran 𝑡 ∈ ran 𝑈 → ∪ ran 𝑡 ⊆ ∪ ran 𝑈) | |
26 | 24, 25 | ax-mp 5 | . 2 ⊢ ∪ ran 𝑡 ⊆ ∪ ran 𝑈 |
27 | 21, 26 | eqssi 3931 | 1 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 ifcif 4425 ∪ cuni 4800 ran crn 5520 Fn wfn 6319 ‘cfv 6324 ∈ cmpo 7137 ωcom 7560 seqωcseqom 8066 |
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-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 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-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 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-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-seqom 8067 |
This theorem is referenced by: fin23lem17 9749 fin23lem31 9754 |
Copyright terms: Public domain | W3C validator |