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Mirrors > Home > MPE Home > Th. List > fin23lem16 | Structured version Visualization version GIF version |
Description: Lemma for fin23 9094. 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 4405 | . . 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 7437 | . . . . 5 ⊢ 𝑈 Fn ω |
4 | fvelrnb 6153 | . . . . 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 6977 | . . . . . . . 8 ⊢ ∅ ∈ ω | |
7 | 0ss 3924 | . . . . . . . . 9 ⊢ ∅ ⊆ 𝑏 | |
8 | 2 | fin23lem15 9039 | . . . . . . . . 9 ⊢ (((𝑏 ∈ ω ∧ ∅ ∈ ω) ∧ ∅ ⊆ 𝑏) → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
9 | 7, 8 | mpan2 703 | . . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ ∅ ∈ ω) → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
10 | 6, 9 | mpan2 703 | . . . . . . 7 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
11 | vex 3176 | . . . . . . . . . 10 ⊢ 𝑡 ∈ V | |
12 | 11 | rnex 6992 | . . . . . . . . 9 ⊢ ran 𝑡 ∈ V |
13 | 12 | uniex 6851 | . . . . . . . 8 ⊢ ∪ ran 𝑡 ∈ V |
14 | 2 | seqom0g 7438 | . . . . . . . 8 ⊢ (∪ ran 𝑡 ∈ V → (𝑈‘∅) = ∪ ran 𝑡) |
15 | 13, 14 | ax-mp 5 | . . . . . . 7 ⊢ (𝑈‘∅) = ∪ ran 𝑡 |
16 | 10, 15 | syl6sseq 3614 | . . . . . 6 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ ∪ ran 𝑡) |
17 | sseq1 3589 | . . . . . 6 ⊢ ((𝑈‘𝑏) = 𝑎 → ((𝑈‘𝑏) ⊆ ∪ ran 𝑡 ↔ 𝑎 ⊆ ∪ ran 𝑡)) | |
18 | 16, 17 | syl5ibcom 234 | . . . . 5 ⊢ (𝑏 ∈ ω → ((𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡)) |
19 | 18 | rexlimiv 3009 | . . . 4 ⊢ (∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡) |
20 | 5, 19 | sylbi 206 | . . 3 ⊢ (𝑎 ∈ ran 𝑈 → 𝑎 ⊆ ∪ ran 𝑡) |
21 | 1, 20 | mprgbir 2911 | . 2 ⊢ ∪ ran 𝑈 ⊆ ∪ ran 𝑡 |
22 | fnfvelrn 6264 | . . . . 5 ⊢ ((𝑈 Fn ω ∧ ∅ ∈ ω) → (𝑈‘∅) ∈ ran 𝑈) | |
23 | 3, 6, 22 | mp2an 704 | . . . 4 ⊢ (𝑈‘∅) ∈ ran 𝑈 |
24 | 15, 23 | eqeltrri 2685 | . . 3 ⊢ ∪ ran 𝑡 ∈ ran 𝑈 |
25 | elssuni 4403 | . . 3 ⊢ (∪ ran 𝑡 ∈ ran 𝑈 → ∪ ran 𝑡 ⊆ ∪ ran 𝑈) | |
26 | 24, 25 | ax-mp 5 | . 2 ⊢ ∪ ran 𝑡 ⊆ ∪ ran 𝑈 |
27 | 21, 26 | eqssi 3584 | 1 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ifcif 4036 ∪ cuni 4372 ran crn 5039 Fn wfn 5799 ‘cfv 5804 ↦ cmpt2 6551 ωcom 6957 seq𝜔cseqom 7429 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 |
This theorem is referenced by: fin23lem17 9043 fin23lem31 9048 |
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