![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pmtrdifellem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for pmtrdifel 18283. (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 |
---|---|
pmtrdifellem2 | ⊢ (𝑄 ∈ 𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtrdifel.0 | . . . 4 ⊢ 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) | |
2 | 1 | difeq1i 3947 | . . 3 ⊢ (𝑆 ∖ I ) = (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I ) |
3 | 2 | dmeqi 5570 | . 2 ⊢ dom (𝑆 ∖ I ) = dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I ) |
4 | eqid 2778 | . . . . 5 ⊢ (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾})) | |
5 | pmtrdifel.t | . . . . 5 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) | |
6 | 4, 5 | pmtrfb 18268 | . . . 4 ⊢ (𝑄 ∈ 𝑇 ↔ ((𝑁 ∖ {𝐾}) ∈ V ∧ 𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ∧ dom (𝑄 ∖ I ) ≈ 2o)) |
7 | difsnexi 7247 | . . . . 5 ⊢ ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V) | |
8 | f1of 6391 | . . . . . 6 ⊢ (𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → 𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾})) | |
9 | fdm 6299 | . . . . . 6 ⊢ (𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) → dom 𝑄 = (𝑁 ∖ {𝐾})) | |
10 | difssd 3961 | . . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑄 ∖ I ) ⊆ 𝑄) | |
11 | dmss 5568 | . . . . . . . 8 ⊢ ((𝑄 ∖ I ) ⊆ 𝑄 → dom (𝑄 ∖ I ) ⊆ dom 𝑄) | |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ dom 𝑄) |
13 | difssd 3961 | . . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑁 ∖ {𝐾}) ⊆ 𝑁) | |
14 | sseq1 3845 | . . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (dom 𝑄 ⊆ 𝑁 ↔ (𝑁 ∖ {𝐾}) ⊆ 𝑁)) | |
15 | 13, 14 | mpbird 249 | . . . . . . 7 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom 𝑄 ⊆ 𝑁) |
16 | 12, 15 | sstrd 3831 | . . . . . 6 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ 𝑁) |
17 | 8, 9, 16 | 3syl 18 | . . . . 5 ⊢ (𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ 𝑁) |
18 | id 22 | . . . . 5 ⊢ (dom (𝑄 ∖ I ) ≈ 2o → dom (𝑄 ∖ I ) ≈ 2o) | |
19 | 7, 17, 18 | 3anim123i 1151 | . . . 4 ⊢ (((𝑁 ∖ {𝐾}) ∈ V ∧ 𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ∧ dom (𝑄 ∖ I ) ≈ 2o) → (𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2o)) |
20 | 6, 19 | sylbi 209 | . . 3 ⊢ (𝑄 ∈ 𝑇 → (𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2o)) |
21 | eqid 2778 | . . . 4 ⊢ (pmTrsp‘𝑁) = (pmTrsp‘𝑁) | |
22 | 21 | pmtrmvd 18259 | . . 3 ⊢ ((𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2o) → dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I ) = dom (𝑄 ∖ I )) |
23 | 20, 22 | syl 17 | . 2 ⊢ (𝑄 ∈ 𝑇 → dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I ) = dom (𝑄 ∖ I )) |
24 | 3, 23 | syl5eq 2826 | 1 ⊢ (𝑄 ∈ 𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∖ cdif 3789 ⊆ wss 3792 {csn 4398 class class class wbr 4886 I cid 5260 dom cdm 5355 ran crn 5356 ⟶wf 6131 –1-1-onto→wf1o 6134 ‘cfv 6135 2oc2o 7837 ≈ cen 8238 pmTrspcpmtr 18244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-om 7344 df-1o 7843 df-2o 7844 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pmtr 18245 |
This theorem is referenced by: pmtrdifellem3 18281 pmtrdifellem4 18282 |
Copyright terms: Public domain | W3C validator |