| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > isf32lem10 | Structured version Visualization version GIF version | ||
| Description: Lemma for isfin3-2 . Write in terms of weak dominance. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
| Ref | Expression |
|---|---|
| isf32lem.a | ⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) |
| isf32lem.b | ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) |
| isf32lem.c | ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) |
| isf32lem.d | ⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} |
| isf32lem.e | ⊢ 𝐽 = (𝑢 ∈ ω ↦ (℩𝑣 ∈ 𝑆 (𝑣 ∩ 𝑆) ≈ 𝑢)) |
| isf32lem.f | ⊢ 𝐾 = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽) |
| isf32lem.g | ⊢ 𝐿 = (𝑡 ∈ 𝐺 ↦ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)))) |
| Ref | Expression |
|---|---|
| isf32lem10 | ⊢ (𝜑 → (𝐺 ∈ 𝑉 → ω ≼* 𝐺)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | isf32lem.a | . . 3 ⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) | |
| 2 | isf32lem.b | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) | |
| 3 | isf32lem.c | . . 3 ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) | |
| 4 | isf32lem.d | . . 3 ⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} | |
| 5 | isf32lem.e | . . 3 ⊢ 𝐽 = (𝑢 ∈ ω ↦ (℩𝑣 ∈ 𝑆 (𝑣 ∩ 𝑆) ≈ 𝑢)) | |
| 6 | isf32lem.f | . . 3 ⊢ 𝐾 = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽) | |
| 7 | isf32lem.g | . . 3 ⊢ 𝐿 = (𝑡 ∈ 𝐺 ↦ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾‘𝑠)))) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | isf32lem9 10401 | . 2 ⊢ (𝜑 → 𝐿:𝐺–onto→ω) |
| 9 | fof 6820 | . . . . 5 ⊢ (𝐿:𝐺–onto→ω → 𝐿:𝐺⟶ω) | |
| 10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿:𝐺⟶ω) |
| 11 | fex 7246 | . . . 4 ⊢ ((𝐿:𝐺⟶ω ∧ 𝐺 ∈ 𝑉) → 𝐿 ∈ V) | |
| 12 | 10, 11 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝑉) → 𝐿 ∈ V) |
| 13 | 12 | ex 412 | . 2 ⊢ (𝜑 → (𝐺 ∈ 𝑉 → 𝐿 ∈ V)) |
| 14 | fowdom 9611 | . . 3 ⊢ ((𝐿 ∈ V ∧ 𝐿:𝐺–onto→ω) → ω ≼* 𝐺) | |
| 15 | 14 | expcom 413 | . 2 ⊢ (𝐿:𝐺–onto→ω → (𝐿 ∈ V → ω ≼* 𝐺)) |
| 16 | 8, 13, 15 | sylsyld 61 | 1 ⊢ (𝜑 → (𝐺 ∈ 𝑉 → ω ≼* 𝐺)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ⊊ wpss 3952 𝒫 cpw 4600 ∩ cint 4946 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ∘ ccom 5689 suc csuc 6386 ℩cio 6512 ⟶wf 6557 –onto→wfo 6559 ‘cfv 6561 ℩crio 7387 ωcom 7887 ≈ cen 8982 ≼* cwdom 9604 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 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-rmo 3380 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-int 4947 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-se 5638 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-isom 6570 df-riota 7388 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-wdom 9605 df-card 9979 |
| This theorem is referenced by: isf32lem11 10403 |
| Copyright terms: Public domain | W3C validator |