| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > fin23lem16 | Structured version Visualization version GIF version | ||
| Description: Lemma for fin23 10429. 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 4939 | . . 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 8495 | . . . . 5 ⊢ 𝑈 Fn ω |
| 4 | fvelrnb 6969 | . . . . 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 7910 | . . . . . . . 8 ⊢ ∅ ∈ ω | |
| 7 | 0ss 4400 | . . . . . . . . 9 ⊢ ∅ ⊆ 𝑏 | |
| 8 | 2 | fin23lem15 10374 | . . . . . . . . 9 ⊢ (((𝑏 ∈ ω ∧ ∅ ∈ ω) ∧ ∅ ⊆ 𝑏) → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
| 9 | 7, 8 | mpan2 691 | . . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ ∅ ∈ ω) → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
| 10 | 6, 9 | mpan2 691 | . . . . . . 7 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ (𝑈‘∅)) |
| 11 | vex 3484 | . . . . . . . . . 10 ⊢ 𝑡 ∈ V | |
| 12 | 11 | rnex 7932 | . . . . . . . . 9 ⊢ ran 𝑡 ∈ V |
| 13 | 12 | uniex 7761 | . . . . . . . 8 ⊢ ∪ ran 𝑡 ∈ V |
| 14 | 2 | seqom0g 8496 | . . . . . . . 8 ⊢ (∪ ran 𝑡 ∈ V → (𝑈‘∅) = ∪ ran 𝑡) |
| 15 | 13, 14 | ax-mp 5 | . . . . . . 7 ⊢ (𝑈‘∅) = ∪ ran 𝑡 |
| 16 | 10, 15 | sseqtrdi 4024 | . . . . . 6 ⊢ (𝑏 ∈ ω → (𝑈‘𝑏) ⊆ ∪ ran 𝑡) |
| 17 | sseq1 4009 | . . . . . 6 ⊢ ((𝑈‘𝑏) = 𝑎 → ((𝑈‘𝑏) ⊆ ∪ ran 𝑡 ↔ 𝑎 ⊆ ∪ ran 𝑡)) | |
| 18 | 16, 17 | syl5ibcom 245 | . . . . 5 ⊢ (𝑏 ∈ ω → ((𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡)) |
| 19 | 18 | rexlimiv 3148 | . . . 4 ⊢ (∃𝑏 ∈ ω (𝑈‘𝑏) = 𝑎 → 𝑎 ⊆ ∪ ran 𝑡) |
| 20 | 5, 19 | sylbi 217 | . . 3 ⊢ (𝑎 ∈ ran 𝑈 → 𝑎 ⊆ ∪ ran 𝑡) |
| 21 | 1, 20 | mprgbir 3068 | . 2 ⊢ ∪ ran 𝑈 ⊆ ∪ ran 𝑡 |
| 22 | fnfvelrn 7100 | . . . . 5 ⊢ ((𝑈 Fn ω ∧ ∅ ∈ ω) → (𝑈‘∅) ∈ ran 𝑈) | |
| 23 | 3, 6, 22 | mp2an 692 | . . . 4 ⊢ (𝑈‘∅) ∈ ran 𝑈 |
| 24 | 15, 23 | eqeltrri 2838 | . . 3 ⊢ ∪ ran 𝑡 ∈ ran 𝑈 |
| 25 | elssuni 4937 | . . 3 ⊢ (∪ ran 𝑡 ∈ ran 𝑈 → ∪ ran 𝑡 ⊆ ∪ ran 𝑈) | |
| 26 | 24, 25 | ax-mp 5 | . 2 ⊢ ∪ ran 𝑡 ⊆ ∪ ran 𝑈 |
| 27 | 21, 26 | eqssi 4000 | 1 ⊢ ∪ ran 𝑈 = ∪ ran 𝑡 |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ifcif 4525 ∪ cuni 4907 ran crn 5686 Fn wfn 6556 ‘cfv 6561 ∈ cmpo 7433 ωcom 7887 seqωcseqom 8487 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seqom 8488 |
| This theorem is referenced by: fin23lem17 10378 fin23lem31 10383 |
| Copyright terms: Public domain | W3C validator |