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Mirrors > Home > MPE Home > Th. List > pmtrdifel | Structured version Visualization version GIF version |
Description: A transposition of elements of a set without a special element corresponds to a transposition of elements of the set. (Contributed by AV, 15-Jan-2019.) |
Ref | Expression |
---|---|
pmtrdifel.t | ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) |
pmtrdifel.r | ⊢ 𝑅 = ran (pmTrsp‘𝑁) |
Ref | Expression |
---|---|
pmtrdifel | ⊢ ∀𝑡 ∈ 𝑇 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (𝑟‘𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtrdifel.t | . . . 4 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾})) | |
2 | pmtrdifel.r | . . . 4 ⊢ 𝑅 = ran (pmTrsp‘𝑁) | |
3 | eqid 2739 | . . . 4 ⊢ ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) | |
4 | 1, 2, 3 | pmtrdifellem1 18898 | . . 3 ⊢ (𝑡 ∈ 𝑇 → ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) ∈ 𝑅) |
5 | 1, 2, 3 | pmtrdifellem3 18900 | . . 3 ⊢ (𝑡 ∈ 𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)) |
6 | fveq1 6737 | . . . . . 6 ⊢ (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → (𝑟‘𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)) | |
7 | 6 | eqeq2d 2750 | . . . . 5 ⊢ (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → ((𝑡‘𝑥) = (𝑟‘𝑥) ↔ (𝑡‘𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥))) |
8 | 7 | ralbidv 3120 | . . . 4 ⊢ (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → (∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (𝑟‘𝑥) ↔ ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥))) |
9 | 8 | rspcev 3551 | . . 3 ⊢ ((((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) ∈ 𝑅 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)) → ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (𝑟‘𝑥)) |
10 | 4, 5, 9 | syl2anc 587 | . 2 ⊢ (𝑡 ∈ 𝑇 → ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (𝑟‘𝑥)) |
11 | 10 | rgen 3073 | 1 ⊢ ∀𝑡 ∈ 𝑇 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (𝑟‘𝑥) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 ∀wral 3063 ∃wrex 3064 ∖ cdif 3879 {csn 4557 I cid 5470 dom cdm 5568 ran crn 5569 ‘cfv 6400 pmTrspcpmtr 18863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3711 df-csb 3828 df-dif 3885 df-un 3887 df-in 3889 df-ss 3899 df-pss 3901 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-om 7666 df-1o 8225 df-2o 8226 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-fin 8653 df-pmtr 18864 |
This theorem is referenced by: (None) |
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