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Mirrors > Home > MPE Home > Th. List > isf32lem8 | Structured version Visualization version GIF version |
Description: Lemma for isfin3-2 9778. K sets are subsets of the base. (Contributed by Stefan O'Rear, 6-Nov-2014.) |
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) |
Ref | Expression |
---|---|
isf32lem8 | ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐾‘𝐴) ⊆ 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isf32lem.f | . . . 4 ⊢ 𝐾 = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽) | |
2 | 1 | fveq1i 6646 | . . 3 ⊢ (𝐾‘𝐴) = (((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)‘𝐴) |
3 | isf32lem.d | . . . . . . . 8 ⊢ 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹‘𝑦)} | |
4 | 3 | ssrab3 4008 | . . . . . . 7 ⊢ 𝑆 ⊆ ω |
5 | isf32lem.a | . . . . . . . 8 ⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺) | |
6 | isf32lem.b | . . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) | |
7 | isf32lem.c | . . . . . . . 8 ⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹) | |
8 | 5, 6, 7, 3 | isf32lem5 9768 | . . . . . . 7 ⊢ (𝜑 → ¬ 𝑆 ∈ Fin) |
9 | isf32lem.e | . . . . . . . 8 ⊢ 𝐽 = (𝑢 ∈ ω ↦ (℩𝑣 ∈ 𝑆 (𝑣 ∩ 𝑆) ≈ 𝑢)) | |
10 | 9 | fin23lem22 9738 | . . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐽:ω–1-1-onto→𝑆) |
11 | 4, 8, 10 | sylancr 590 | . . . . . 6 ⊢ (𝜑 → 𝐽:ω–1-1-onto→𝑆) |
12 | f1of 6590 | . . . . . 6 ⊢ (𝐽:ω–1-1-onto→𝑆 → 𝐽:ω⟶𝑆) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽:ω⟶𝑆) |
14 | fvco3 6737 | . . . . 5 ⊢ ((𝐽:ω⟶𝑆 ∧ 𝐴 ∈ ω) → (((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)‘𝐴) = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤)))‘(𝐽‘𝐴))) | |
15 | 13, 14 | sylan 583 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)‘𝐴) = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤)))‘(𝐽‘𝐴))) |
16 | 13 | ffvelrnda 6828 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐽‘𝐴) ∈ 𝑆) |
17 | fveq2 6645 | . . . . . . 7 ⊢ (𝑤 = (𝐽‘𝐴) → (𝐹‘𝑤) = (𝐹‘(𝐽‘𝐴))) | |
18 | suceq 6224 | . . . . . . . 8 ⊢ (𝑤 = (𝐽‘𝐴) → suc 𝑤 = suc (𝐽‘𝐴)) | |
19 | 18 | fveq2d 6649 | . . . . . . 7 ⊢ (𝑤 = (𝐽‘𝐴) → (𝐹‘suc 𝑤) = (𝐹‘suc (𝐽‘𝐴))) |
20 | 17, 19 | difeq12d 4051 | . . . . . 6 ⊢ (𝑤 = (𝐽‘𝐴) → ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤)) = ((𝐹‘(𝐽‘𝐴)) ∖ (𝐹‘suc (𝐽‘𝐴)))) |
21 | eqid 2798 | . . . . . 6 ⊢ (𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) = (𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) | |
22 | fvex 6658 | . . . . . . 7 ⊢ (𝐹‘(𝐽‘𝐴)) ∈ V | |
23 | 22 | difexi 5196 | . . . . . 6 ⊢ ((𝐹‘(𝐽‘𝐴)) ∖ (𝐹‘suc (𝐽‘𝐴))) ∈ V |
24 | 20, 21, 23 | fvmpt 6745 | . . . . 5 ⊢ ((𝐽‘𝐴) ∈ 𝑆 → ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤)))‘(𝐽‘𝐴)) = ((𝐹‘(𝐽‘𝐴)) ∖ (𝐹‘suc (𝐽‘𝐴)))) |
25 | 16, 24 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤)))‘(𝐽‘𝐴)) = ((𝐹‘(𝐽‘𝐴)) ∖ (𝐹‘suc (𝐽‘𝐴)))) |
26 | 15, 25 | eqtrd 2833 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)‘𝐴) = ((𝐹‘(𝐽‘𝐴)) ∖ (𝐹‘suc (𝐽‘𝐴)))) |
27 | 2, 26 | syl5eq 2845 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐾‘𝐴) = ((𝐹‘(𝐽‘𝐴)) ∖ (𝐹‘suc (𝐽‘𝐴)))) |
28 | 5 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → 𝐹:ω⟶𝒫 𝐺) |
29 | 4, 16 | sseldi 3913 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐽‘𝐴) ∈ ω) |
30 | 28, 29 | ffvelrnd 6829 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐹‘(𝐽‘𝐴)) ∈ 𝒫 𝐺) |
31 | 30 | elpwid 4508 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐹‘(𝐽‘𝐴)) ⊆ 𝐺) |
32 | 31 | ssdifssd 4070 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝐹‘(𝐽‘𝐴)) ∖ (𝐹‘suc (𝐽‘𝐴))) ⊆ 𝐺) |
33 | 27, 32 | eqsstrd 3953 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐾‘𝐴) ⊆ 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ⊊ wpss 3882 𝒫 cpw 4497 ∩ cint 4838 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 ∘ ccom 5523 suc csuc 6161 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 ℩crio 7092 ωcom 7560 ≈ cen 8489 Fincfn 8492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-om 7561 df-wrecs 7930 df-recs 7991 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 |
This theorem is referenced by: isf32lem9 9772 |
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