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Mirrors > Home > MPE Home > Th. List > unblem4 | Structured version Visualization version GIF version |
Description: Lemma for unbnn 9298. The function 𝐹 maps the set of natural numbers one-to-one to the set of unbounded natural numbers 𝐴. (Contributed by NM, 3-Dec-2003.) |
Ref | Expression |
---|---|
unblem.2 | ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω) |
Ref | Expression |
---|---|
unblem4 | ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → 𝐹:ω–1-1→𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omsson 7855 | . . . 4 ⊢ ω ⊆ On | |
2 | sstr 3985 | . . . 4 ⊢ ((𝐴 ⊆ ω ∧ ω ⊆ On) → 𝐴 ⊆ On) | |
3 | 1, 2 | mpan2 688 | . . 3 ⊢ (𝐴 ⊆ ω → 𝐴 ⊆ On) |
4 | 3 | adantr 480 | . 2 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → 𝐴 ⊆ On) |
5 | frfnom 8433 | . . . 4 ⊢ (rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω) Fn ω | |
6 | unblem.2 | . . . . 5 ⊢ 𝐹 = (rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω) | |
7 | 6 | fneq1i 6639 | . . . 4 ⊢ (𝐹 Fn ω ↔ (rec((𝑥 ∈ V ↦ ∩ (𝐴 ∖ suc 𝑥)), ∩ 𝐴) ↾ ω) Fn ω) |
8 | 5, 7 | mpbir 230 | . . 3 ⊢ 𝐹 Fn ω |
9 | 6 | unblem2 9295 | . . . 4 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ 𝐴)) |
10 | 9 | ralrimiv 3139 | . . 3 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → ∀𝑧 ∈ ω (𝐹‘𝑧) ∈ 𝐴) |
11 | ffnfv 7113 | . . . 4 ⊢ (𝐹:ω⟶𝐴 ↔ (𝐹 Fn ω ∧ ∀𝑧 ∈ ω (𝐹‘𝑧) ∈ 𝐴)) | |
12 | 11 | biimpri 227 | . . 3 ⊢ ((𝐹 Fn ω ∧ ∀𝑧 ∈ ω (𝐹‘𝑧) ∈ 𝐴) → 𝐹:ω⟶𝐴) |
13 | 8, 10, 12 | sylancr 586 | . 2 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → 𝐹:ω⟶𝐴) |
14 | 6 | unblem3 9296 | . . 3 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → (𝑧 ∈ ω → (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧))) |
15 | 14 | ralrimiv 3139 | . 2 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → ∀𝑧 ∈ ω (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧)) |
16 | omsmo 8656 | . 2 ⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑧 ∈ ω (𝐹‘𝑧) ∈ (𝐹‘suc 𝑧)) → 𝐹:ω–1-1→𝐴) | |
17 | 4, 13, 15, 16 | syl21anc 835 | 1 ⊢ ((𝐴 ⊆ ω ∧ ∀𝑤 ∈ ω ∃𝑣 ∈ 𝐴 𝑤 ∈ 𝑣) → 𝐹:ω–1-1→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 Vcvv 3468 ∖ cdif 3940 ⊆ wss 3943 ∩ cint 4943 ↦ cmpt 5224 ↾ cres 5671 Oncon0 6357 suc csuc 6359 Fn wfn 6531 ⟶wf 6532 –1-1→wf1 6533 ‘cfv 6536 ωcom 7851 reccrdg 8407 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 |
This theorem is referenced by: unbnn 9298 |
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