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Mirrors > Home > MPE Home > Th. List > fmfnfmlem1 | Structured version Visualization version GIF version |
Description: Lemma for fmfnfm 21572. (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 21451 | . . . . 5 ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑠 ∈ (fi‘𝐵)) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ 𝑠) | |
3 | 1, 2 | sylan 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (fi‘𝐵)) → ∃𝑤 ∈ 𝐵 𝑤 ⊆ 𝑠) |
4 | imass2 5420 | . . . . . . 7 ⊢ (𝑤 ⊆ 𝑠 → (𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑠)) | |
5 | sstr2 3575 | . . . . . . 7 ⊢ ((𝐹 “ 𝑤) ⊆ (𝐹 “ 𝑠) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝐹 “ 𝑤) ⊆ 𝑡)) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ (𝑤 ⊆ 𝑠 → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝐹 “ 𝑤) ⊆ 𝑡)) |
7 | 6 | com12 32 | . . . . 5 ⊢ ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑤 ⊆ 𝑠 → (𝐹 “ 𝑤) ⊆ 𝑡)) |
8 | 7 | reximdv 2999 | . . . 4 ⊢ ((𝐹 “ 𝑠) ⊆ 𝑡 → (∃𝑤 ∈ 𝐵 𝑤 ⊆ 𝑠 → ∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡)) |
9 | 3, 8 | syl5com 31 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (fi‘𝐵)) → ((𝐹 “ 𝑠) ⊆ 𝑡 → ∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡)) |
10 | fmfnfm.l | . . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋)) | |
11 | filtop 21469 | . . . . . . . 8 ⊢ (𝐿 ∈ (Fil‘𝑋) → 𝑋 ∈ 𝐿) | |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐿) |
13 | fmfnfm.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝑌⟶𝑋) | |
14 | elfm 21561 | . . . . . . 7 ⊢ ((𝑋 ∈ 𝐿 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡))) | |
15 | 12, 1, 13, 14 | syl3anc 1318 | . . . . . 6 ⊢ (𝜑 → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡))) |
16 | fmfnfm.fm | . . . . . . 7 ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿) | |
17 | 16 | sseld 3567 | . . . . . 6 ⊢ (𝜑 → (𝑡 ∈ ((𝑋 FilMap 𝐹)‘𝐵) → 𝑡 ∈ 𝐿)) |
18 | 15, 17 | sylbird 249 | . . . . 5 ⊢ (𝜑 → ((𝑡 ⊆ 𝑋 ∧ ∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡) → 𝑡 ∈ 𝐿)) |
19 | 18 | expcomd 453 | . . . 4 ⊢ (𝜑 → (∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))) |
20 | 19 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (fi‘𝐵)) → (∃𝑤 ∈ 𝐵 (𝐹 “ 𝑤) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))) |
21 | 9, 20 | syld 46 | . 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (fi‘𝐵)) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿))) |
22 | 21 | ex 449 | 1 ⊢ (𝜑 → (𝑠 ∈ (fi‘𝐵) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ficfi 8199 fBascfbas 19555 Filcfil 21459 FilMap cfm 21547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-fin 7845 df-fi 8200 df-fbas 19564 df-fg 19565 df-fil 21460 df-fm 21552 |
This theorem is referenced by: fmfnfmlem4 21571 |
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