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Mirrors > Home > MPE Home > Th. List > fmfnfmlem1 | Structured version Visualization version GIF version |
Description: Lemma for fmfnfm 21762. (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 21641 | . . . . 5 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠 ∈ (fi‘𝐵)) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ 𝑠) | |
3 | 1, 2 | sylan 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (fi‘𝐵)) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ 𝑠) |
4 | sstr2 3610 | . . . . . 6 ⊢ ((𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑠) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝐹 “ 𝑤) ⊆ 𝑡)) | |
5 | imass2 5501 | . . . . . 6 ⊢ (𝑤 ⊆ 𝑠 → (𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑠)) | |
6 | 4, 5 | syl11 33 | . . . . 5 ⊢ ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑤 ⊆ 𝑠 → (𝐹 “ 𝑤) ⊆ 𝑡)) |
7 | 6 | reximdv 3016 | . . . 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 21659 | . . . . . . . 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 21751 | . . . . . . 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 1326 | . . . . . 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 3602 | . . . . . 6 ⊢ (𝜑 → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘𝐵) → 𝑡 ∈ 𝐿)) |
17 | 14, 16 | sylbird 250 | . . . . 5 ⊢ (𝜑 → ((𝑡 ⊆ 𝑋 ∧ ∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡) → 𝑡 ∈ 𝐿)) |
18 | 17 | expcomd 454 | . . . 4 ⊢ (𝜑 → (∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))) |
19 | 18 | adantr 481 | . . 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 450 | 1 ⊢ (𝜑 → (𝑠 ∈ (fi‘𝐵) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∈ wcel 1990 ∃wrex 2913 ⊆ wss 3574 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ficfi 8316 fBascfbas 19734 Filcfil 21649 FilMap cfm 21737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-fin 7959 df-fi 8317 df-fbas 19743 df-fg 19744 df-fil 21650 df-fm 21742 |
This theorem is referenced by: fmfnfmlem4 21761 |
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