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Mirrors > Home > MPE Home > Th. List > isf32lem11 | Structured version Visualization version GIF version |
Description: Lemma for isfin3-2 9789. Remove hypotheses from isf32lem10 9784. (Contributed by Stefan O'Rear, 17-May-2015.) |
Ref | Expression |
---|---|
isf32lem11 | ⊢ ((𝐺 ∈ 𝑉 ∧ (𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹)) → ω ≼* 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . 3 ⊢ ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹) → 𝐹:ω⟶𝒫 𝐺) | |
2 | suceq 6256 | . . . . . . . 8 ⊢ (𝑏 = 𝑐 → suc 𝑏 = suc 𝑐) | |
3 | 2 | fveq2d 6674 | . . . . . . 7 ⊢ (𝑏 = 𝑐 → (𝐹‘suc 𝑏) = (𝐹‘suc 𝑐)) |
4 | fveq2 6670 | . . . . . . 7 ⊢ (𝑏 = 𝑐 → (𝐹‘𝑏) = (𝐹‘𝑐)) | |
5 | 3, 4 | sseq12d 4000 | . . . . . 6 ⊢ (𝑏 = 𝑐 → ((𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ↔ (𝐹‘suc 𝑐) ⊆ (𝐹‘𝑐))) |
6 | 5 | cbvralvw 3449 | . . . . 5 ⊢ (∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ↔ ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹‘𝑐)) |
7 | 6 | biimpi 218 | . . . 4 ⊢ (∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) → ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹‘𝑐)) |
8 | 7 | 3ad2ant2 1130 | . . 3 ⊢ ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹) → ∀𝑐 ∈ ω (𝐹‘suc 𝑐) ⊆ (𝐹‘𝑐)) |
9 | simp3 1134 | . . 3 ⊢ ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹) → ¬ ∩ ran 𝐹 ∈ ran 𝐹) | |
10 | suceq 6256 | . . . . . 6 ⊢ (𝑒 = 𝑑 → suc 𝑒 = suc 𝑑) | |
11 | 10 | fveq2d 6674 | . . . . 5 ⊢ (𝑒 = 𝑑 → (𝐹‘suc 𝑒) = (𝐹‘suc 𝑑)) |
12 | fveq2 6670 | . . . . 5 ⊢ (𝑒 = 𝑑 → (𝐹‘𝑒) = (𝐹‘𝑑)) | |
13 | 11, 12 | psseq12d 4071 | . . . 4 ⊢ (𝑒 = 𝑑 → ((𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒) ↔ (𝐹‘suc 𝑑) ⊊ (𝐹‘𝑑))) |
14 | 13 | cbvrabv 3491 | . . 3 ⊢ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} = {𝑑 ∈ ω ∣ (𝐹‘suc 𝑑) ⊊ (𝐹‘𝑑)} |
15 | eqid 2821 | . . 3 ⊢ (𝑓 ∈ ω ↦ (℩𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)}) ≈ 𝑓)) = (𝑓 ∈ ω ↦ (℩𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)}) ≈ 𝑓)) | |
16 | eqid 2821 | . . 3 ⊢ ((ℎ ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} ↦ ((𝐹‘ℎ) ∖ (𝐹‘suc ℎ))) ∘ (𝑓 ∈ ω ↦ (℩𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)}) ≈ 𝑓))) = ((ℎ ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} ↦ ((𝐹‘ℎ) ∖ (𝐹‘suc ℎ))) ∘ (𝑓 ∈ ω ↦ (℩𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)}) ≈ 𝑓))) | |
17 | eqid 2821 | . . 3 ⊢ (𝑘 ∈ 𝐺 ↦ (℩𝑙(𝑙 ∈ ω ∧ 𝑘 ∈ (((ℎ ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} ↦ ((𝐹‘ℎ) ∖ (𝐹‘suc ℎ))) ∘ (𝑓 ∈ ω ↦ (℩𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)}) ≈ 𝑓)))‘𝑙)))) = (𝑘 ∈ 𝐺 ↦ (℩𝑙(𝑙 ∈ ω ∧ 𝑘 ∈ (((ℎ ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} ↦ ((𝐹‘ℎ) ∖ (𝐹‘suc ℎ))) ∘ (𝑓 ∈ ω ↦ (℩𝑔 ∈ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)} (𝑔 ∩ {𝑒 ∈ ω ∣ (𝐹‘suc 𝑒) ⊊ (𝐹‘𝑒)}) ≈ 𝑓)))‘𝑙)))) | |
18 | 1, 8, 9, 14, 15, 16, 17 | isf32lem10 9784 | . 2 ⊢ ((𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹) → (𝐺 ∈ 𝑉 → ω ≼* 𝐺)) |
19 | 18 | impcom 410 | 1 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝐹:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹)) → ω ≼* 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2114 ∀wral 3138 {crab 3142 ∖ 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 ‘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: isf32lem12 9786 fin33i 9791 |
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