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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 9066 | . 2 ⊢ (𝜑 → 𝐿:𝐺–onto→ω) |
9 | fof 6028 | . . . . 5 ⊢ (𝐿:𝐺–onto→ω → 𝐿:𝐺⟶ω) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿:𝐺⟶ω) |
11 | fex 6394 | . . . 4 ⊢ ((𝐿:𝐺⟶ω ∧ 𝐺 ∈ 𝑉) → 𝐿 ∈ V) | |
12 | 10, 11 | sylan 487 | . . 3 ⊢ ((𝜑 ∧ 𝐺 ∈ 𝑉) → 𝐿 ∈ V) |
13 | 12 | ex 449 | . 2 ⊢ (𝜑 → (𝐺 ∈ 𝑉 → 𝐿 ∈ V)) |
14 | fowdom 8359 | . . 3 ⊢ ((𝐿 ∈ V ∧ 𝐿:𝐺–onto→ω) → ω ≼* 𝐺) | |
15 | 14 | expcom 450 | . 2 ⊢ (𝐿:𝐺–onto→ω → (𝐿 ∈ V → ω ≼* 𝐺)) |
16 | 8, 13, 15 | sylsyld 59 | 1 ⊢ (𝜑 → (𝐺 ∈ 𝑉 → ω ≼* 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 ⊊ wpss 3541 𝒫 cpw 4108 ∩ cint 4410 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ∘ ccom 5042 suc csuc 5642 ℩cio 5766 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 ℩crio 6510 ωcom 6957 ≈ cen 7838 ≼* cwdom 8345 |
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-rep 4699 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-rmo 2904 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-int 4411 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-se 4998 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-isom 5813 df-riota 6511 df-om 6958 df-wrecs 7294 df-recs 7355 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-wdom 8347 df-card 8648 |
This theorem is referenced by: isf32lem11 9068 |
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