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| Mirrors > Home > MPE Home > Th. List > isf32lem8 | Structured version Visualization version GIF version | ||
| Description: Lemma for isfin3-2 10407. 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 6907 | . . 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 4082 | . . . . . . 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 10397 | . . . . . . 7 ⊢ (𝜑 → ¬ 𝑆 ∈ Fin) |
| 9 | isf32lem.e | . . . . . . . 8 ⊢ 𝐽 = (𝑢 ∈ ω ↦ (℩𝑣 ∈ 𝑆 (𝑣 ∩ 𝑆) ≈ 𝑢)) | |
| 10 | 9 | fin23lem22 10367 | . . . . . . 7 ⊢ ((𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin) → 𝐽:ω–1-1-onto→𝑆) |
| 11 | 4, 8, 10 | sylancr 587 | . . . . . 6 ⊢ (𝜑 → 𝐽:ω–1-1-onto→𝑆) |
| 12 | f1of 6848 | . . . . . 6 ⊢ (𝐽:ω–1-1-onto→𝑆 → 𝐽:ω⟶𝑆) | |
| 13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽:ω⟶𝑆) |
| 14 | fvco3 7008 | . . . . 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 580 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)‘𝐴) = ((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤)))‘(𝐽‘𝐴))) |
| 16 | 13 | ffvelcdmda 7104 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐽‘𝐴) ∈ 𝑆) |
| 17 | fveq2 6906 | . . . . . . 7 ⊢ (𝑤 = (𝐽‘𝐴) → (𝐹‘𝑤) = (𝐹‘(𝐽‘𝐴))) | |
| 18 | suceq 6450 | . . . . . . . 8 ⊢ (𝑤 = (𝐽‘𝐴) → suc 𝑤 = suc (𝐽‘𝐴)) | |
| 19 | 18 | fveq2d 6910 | . . . . . . 7 ⊢ (𝑤 = (𝐽‘𝐴) → (𝐹‘suc 𝑤) = (𝐹‘suc (𝐽‘𝐴))) |
| 20 | 17, 19 | difeq12d 4127 | . . . . . 6 ⊢ (𝑤 = (𝐽‘𝐴) → ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤)) = ((𝐹‘(𝐽‘𝐴)) ∖ (𝐹‘suc (𝐽‘𝐴)))) |
| 21 | eqid 2737 | . . . . . 6 ⊢ (𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) = (𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) | |
| 22 | fvex 6919 | . . . . . . 7 ⊢ (𝐹‘(𝐽‘𝐴)) ∈ V | |
| 23 | 22 | difexi 5330 | . . . . . 6 ⊢ ((𝐹‘(𝐽‘𝐴)) ∖ (𝐹‘suc (𝐽‘𝐴))) ∈ V |
| 24 | 20, 21, 23 | fvmpt 7016 | . . . . 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 2777 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (((𝑤 ∈ 𝑆 ↦ ((𝐹‘𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)‘𝐴) = ((𝐹‘(𝐽‘𝐴)) ∖ (𝐹‘suc (𝐽‘𝐴)))) |
| 27 | 2, 26 | eqtrid 2789 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐾‘𝐴) = ((𝐹‘(𝐽‘𝐴)) ∖ (𝐹‘suc (𝐽‘𝐴)))) |
| 28 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → 𝐹:ω⟶𝒫 𝐺) |
| 29 | 4, 16 | sselid 3981 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐽‘𝐴) ∈ ω) |
| 30 | 28, 29 | ffvelcdmd 7105 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐹‘(𝐽‘𝐴)) ∈ 𝒫 𝐺) |
| 31 | 30 | elpwid 4609 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐹‘(𝐽‘𝐴)) ⊆ 𝐺) |
| 32 | 31 | ssdifssd 4147 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → ((𝐹‘(𝐽‘𝐴)) ∖ (𝐹‘suc (𝐽‘𝐴))) ⊆ 𝐺) |
| 33 | 27, 32 | eqsstrd 4018 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ ω) → (𝐾‘𝐴) ⊆ 𝐺) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ⊊ wpss 3952 𝒫 cpw 4600 ∩ cint 4946 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ∘ ccom 5689 suc csuc 6386 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 ℩crio 7387 ωcom 7887 ≈ cen 8982 Fincfn 8985 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 |
| This theorem is referenced by: isf32lem9 10401 |
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