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Mirrors > Home > MPE Home > Th. List > pmtrdifellem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for pmtrdifel 18359. (Contributed by AV, 15-Jan-2019.) |
Ref | Expression |
---|---|
pmtrdifel.t | ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) |
pmtrdifel.r | ⊢ 𝑅 = ran (pmTrsp‘𝑁) |
pmtrdifel.0 | ⊢ 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) |
Ref | Expression |
---|---|
pmtrdifellem1 | ⊢ (𝑄 ∈ 𝑇 → 𝑆 ∈ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2772 | . . 3 ⊢ (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾})) | |
2 | pmtrdifel.t | . . 3 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) | |
3 | 1, 2 | pmtrfb 18344 | . 2 ⊢ (𝑄 ∈ 𝑇 ↔ ((𝑁 ∖ {𝐾}) ∈ V ∧ 𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ∧ dom (𝑄 ∖ I ) ≈ 2o)) |
4 | difsnexi 7294 | . . 3 ⊢ ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V) | |
5 | f1of 6438 | . . . 4 ⊢ (𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → 𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾})) | |
6 | fdm 6346 | . . . 4 ⊢ (𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) → dom 𝑄 = (𝑁 ∖ {𝐾})) | |
7 | difssd 3995 | . . . . . 6 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑄 ∖ I ) ⊆ 𝑄) | |
8 | dmss 5614 | . . . . . 6 ⊢ ((𝑄 ∖ I ) ⊆ 𝑄 → dom (𝑄 ∖ I ) ⊆ dom 𝑄) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ dom 𝑄) |
10 | difssd 3995 | . . . . . 6 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑁 ∖ {𝐾}) ⊆ 𝑁) | |
11 | sseq1 3878 | . . . . . 6 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (dom 𝑄 ⊆ 𝑁 ↔ (𝑁 ∖ {𝐾}) ⊆ 𝑁)) | |
12 | 10, 11 | mpbird 249 | . . . . 5 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom 𝑄 ⊆ 𝑁) |
13 | 9, 12 | sstrd 3864 | . . . 4 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ 𝑁) |
14 | 5, 6, 13 | 3syl 18 | . . 3 ⊢ (𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ 𝑁) |
15 | id 22 | . . 3 ⊢ (dom (𝑄 ∖ I ) ≈ 2o → dom (𝑄 ∖ I ) ≈ 2o) | |
16 | pmtrdifel.0 | . . . 4 ⊢ 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) | |
17 | eqid 2772 | . . . . 5 ⊢ (pmTrsp‘𝑁) = (pmTrsp‘𝑁) | |
18 | pmtrdifel.r | . . . . 5 ⊢ 𝑅 = ran (pmTrsp‘𝑁) | |
19 | 17, 18 | pmtrrn 18336 | . . . 4 ⊢ ((𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2o) → ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∈ 𝑅) |
20 | 16, 19 | syl5eqel 2864 | . . 3 ⊢ ((𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2o) → 𝑆 ∈ 𝑅) |
21 | 4, 14, 15, 20 | syl3an 1140 | . 2 ⊢ (((𝑁 ∖ {𝐾}) ∈ V ∧ 𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ∧ dom (𝑄 ∖ I ) ≈ 2o) → 𝑆 ∈ 𝑅) |
22 | 3, 21 | sylbi 209 | 1 ⊢ (𝑄 ∈ 𝑇 → 𝑆 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 Vcvv 3409 ∖ cdif 3822 ⊆ wss 3825 {csn 4435 class class class wbr 4923 I cid 5304 dom cdm 5400 ran crn 5401 ⟶wf 6178 –1-1-onto→wf1o 6181 ‘cfv 6182 2oc2o 7891 ≈ cen 8295 pmTrspcpmtr 18320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-om 7391 df-1o 7897 df-2o 7898 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-pmtr 18321 |
This theorem is referenced by: pmtrdifellem3 18357 pmtrdifellem4 18358 pmtrdifel 18359 pmtrdifwrdellem1 18360 pmtrdifwrdellem2 18361 |
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