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 9783 | . 2 ⊢ (𝜑 → 𝐿:𝐺–onto→ω) |
9 | fof 6590 | . . . . 5 ⊢ (𝐿:𝐺–onto→ω → 𝐿:𝐺⟶ω) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿:𝐺⟶ω) |
11 | fex 6989 | . . . 4 ⊢ ((𝐿:𝐺⟶ω ∧ 𝐺 ∈ 𝑉) → 𝐿 ∈ V) | |
12 | 10, 11 | sylan 582 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝑉) → 𝐿 ∈ V) |
13 | 12 | ex 415 | . 2 ⊢ (𝜑 → (𝐺 ∈ 𝑉 → 𝐿 ∈ V)) |
14 | fowdom 9035 | . . 3 ⊢ ((𝐿 ∈ V ∧ 𝐿:𝐺–onto→ω) → ω ≼* 𝐺) | |
15 | 14 | expcom 416 | . 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 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 Vcvv 3494 ∖ cdif 3933 ∩ cin 3935 ⊆ wss 3936 ⊊ wpss 3937 𝒫 cpw 4539 ∩ cint 4876 class class class wbr 5066 ↦ cmpt 5146 ran crn 5556 ∘ ccom 5559 suc csuc 6193 ℩cio 6312 ⟶wf 6351 –onto→wfo 6353 ‘cfv 6355 ℩crio 7113 ωcom 7580 ≈ cen 8506 ≼* cwdom 9021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-om 7581 df-wrecs 7947 df-recs 8008 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-wdom 9023 df-card 9368 |
This theorem is referenced by: isf32lem11 9785 |
Copyright terms: Public domain | W3C validator |