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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 10399 | . 2 ⊢ (𝜑 → 𝐿:𝐺–onto→ω) |
9 | fof 6821 | . . . . 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 9609 | . . 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 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 Vcvv 3478 ∖ cdif 3960 ∩ cin 3962 ⊆ wss 3963 ⊊ wpss 3964 𝒫 cpw 4605 ∩ cint 4951 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 ∘ ccom 5693 suc csuc 6388 ℩cio 6514 ⟶wf 6559 –onto→wfo 6561 ‘cfv 6563 ℩crio 7387 ωcom 7887 ≈ cen 8981 ≼* cwdom 9602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-wdom 9603 df-card 9977 |
This theorem is referenced by: isf32lem11 10401 |
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