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Mirrors > Home > MPE Home > Th. List > fmfnfmlem1 | Structured version Visualization version GIF version |
Description: Lemma for fmfnfm 22560. (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 22439 | . . . . 5 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠 ∈ (fi‘𝐵)) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ 𝑠) | |
3 | 1, 2 | sylan 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (fi‘𝐵)) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ 𝑠) |
4 | sstr2 3974 | . . . . . 6 ⊢ ((𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑠) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝐹 “ 𝑤) ⊆ 𝑡)) | |
5 | imass2 5960 | . . . . . 6 ⊢ (𝑤 ⊆ 𝑠 → (𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑠)) | |
6 | 4, 5 | syl11 33 | . . . . 5 ⊢ ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑤 ⊆ 𝑠 → (𝐹 “ 𝑤) ⊆ 𝑡)) |
7 | 6 | reximdv 3273 | . . . 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 22457 | . . . . . . . 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 22549 | . . . . . . 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 1367 | . . . . . 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 3966 | . . . . . 6 ⊢ (𝜑 → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘𝐵) → 𝑡 ∈ 𝐿)) |
17 | 14, 16 | sylbird 262 | . . . . 5 ⊢ (𝜑 → ((𝑡 ⊆ 𝑋 ∧ ∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡) → 𝑡 ∈ 𝐿)) |
18 | 17 | expcomd 419 | . . . 4 ⊢ (𝜑 → (∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))) |
19 | 18 | adantr 483 | . . 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 415 | 1 ⊢ (𝜑 → (𝑠 ∈ (fi‘𝐵) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∃wrex 3139 ⊆ wss 3936 “ cima 5553 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ficfi 8868 fBascfbas 20527 Filcfil 22447 FilMap cfm 22535 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-fin 8507 df-fi 8869 df-fbas 20536 df-fg 20537 df-fil 22448 df-fm 22540 |
This theorem is referenced by: fmfnfmlem4 22559 |
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