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Mirrors > Home > MPE Home > Th. List > fmfnfmlem1 | Structured version Visualization version GIF version |
Description: Lemma for fmfnfm 22827. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
Ref | Expression |
---|---|
fmfnfm.b | ⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌)) |
fmfnfm.l | ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋)) |
fmfnfm.f | ⊢ (𝜑 → 𝐹:𝑌⟶𝑋) |
fmfnfm.fm | ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿) |
Ref | Expression |
---|---|
fmfnfmlem1 | ⊢ (𝜑 → (𝑠 ∈ (fi‘𝐵) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmfnfm.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌)) | |
2 | fbssfi 22706 | . . . . 5 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠 ∈ (fi‘𝐵)) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ 𝑠) | |
3 | 1, 2 | sylan 583 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (fi‘𝐵)) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ 𝑠) |
4 | sstr2 3898 | . . . . . 6 ⊢ ((𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑠) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝐹 “ 𝑤) ⊆ 𝑡)) | |
5 | imass2 5959 | . . . . . 6 ⊢ (𝑤 ⊆ 𝑠 → (𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑠)) | |
6 | 4, 5 | syl11 33 | . . . . 5 ⊢ ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑤 ⊆ 𝑠 → (𝐹 “ 𝑤) ⊆ 𝑡)) |
7 | 6 | reximdv 3185 | . . . 4 ⊢ ((𝐹 “ 𝑠) ⊆ 𝑡 → (∃𝑤 ∈ 𝐵 𝑤 ⊆ 𝑠 → ∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡)) |
8 | 3, 7 | syl5com 31 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (fi‘𝐵)) → ((𝐹 “ 𝑠) ⊆ 𝑡 → ∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡)) |
9 | fmfnfm.l | . . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋)) | |
10 | filtop 22724 | . . . . . . . 8 ⊢ (𝐿 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐿) | |
11 | 9, 10 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐿) |
12 | fmfnfm.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑌⟶𝑋) | |
13 | elfm 22816 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡))) | |
14 | 11, 1, 12, 13 | syl3anc 1373 | . . . . . 6 ⊢ (𝜑 → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡))) |
15 | fmfnfm.fm | . . . . . . 7 ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿) | |
16 | 15 | sseld 3890 | . . . . . 6 ⊢ (𝜑 → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘𝐵) → 𝑡 ∈ 𝐿)) |
17 | 14, 16 | sylbird 263 | . . . . 5 ⊢ (𝜑 → ((𝑡 ⊆ 𝑋 ∧ ∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡) → 𝑡 ∈ 𝐿)) |
18 | 17 | expcomd 420 | . . . 4 ⊢ (𝜑 → (∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))) |
19 | 18 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (fi‘𝐵)) → (∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))) |
20 | 8, 19 | syld 47 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (fi‘𝐵)) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))) |
21 | 20 | ex 416 | 1 ⊢ (𝜑 → (𝑠 ∈ (fi‘𝐵) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2110 ∃wrex 3055 ⊆ wss 3857 “ cima 5543 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ficfi 9015 fBascfbas 20323 Filcfil 22714 FilMap cfm 22802 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1o 8191 df-er 8380 df-en 8616 df-fin 8619 df-fi 9016 df-fbas 20332 df-fg 20333 df-fil 22715 df-fm 22807 |
This theorem is referenced by: fmfnfmlem4 22826 |
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