Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > subfacp1lem2b | Structured version Visualization version GIF version |
Description: Lemma for subfacp1 32428. Properties of a bijection on 𝐾 augmented with the two-element flip to get a bijection on 𝐾 ∪ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.) |
Ref | Expression |
---|---|
derang.d | ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) |
subfac.n | ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) |
subfacp1lem.a | ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} |
subfacp1lem1.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
subfacp1lem1.m | ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) |
subfacp1lem1.x | ⊢ 𝑀 ∈ V |
subfacp1lem1.k | ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) |
subfacp1lem2.5 | ⊢ 𝐹 = (𝐺 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) |
subfacp1lem2.6 | ⊢ (𝜑 → 𝐺:𝐾–1-1-onto→𝐾) |
Ref | Expression |
---|---|
subfacp1lem2b | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝐺‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | derang.d | . . . . . 6 ⊢ 𝐷 = (𝑥 ∈ Fin ↦ (♯‘{𝑓 ∣ (𝑓:𝑥–1-1-onto→𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) ≠ 𝑦)})) | |
2 | subfac.n | . . . . . 6 ⊢ 𝑆 = (𝑛 ∈ ℕ0 ↦ (𝐷‘(1...𝑛))) | |
3 | subfacp1lem.a | . . . . . 6 ⊢ 𝐴 = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ ∀𝑦 ∈ (1...(𝑁 + 1))(𝑓‘𝑦) ≠ 𝑦)} | |
4 | subfacp1lem1.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
5 | subfacp1lem1.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (2...(𝑁 + 1))) | |
6 | subfacp1lem1.x | . . . . . 6 ⊢ 𝑀 ∈ V | |
7 | subfacp1lem1.k | . . . . . 6 ⊢ 𝐾 = ((2...(𝑁 + 1)) ∖ {𝑀}) | |
8 | subfacp1lem2.5 | . . . . . 6 ⊢ 𝐹 = (𝐺 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) | |
9 | subfacp1lem2.6 | . . . . . 6 ⊢ (𝜑 → 𝐺:𝐾–1-1-onto→𝐾) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | subfacp1lem2a 32422 | . . . . 5 ⊢ (𝜑 → (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘1) = 𝑀 ∧ (𝐹‘𝑀) = 1)) |
11 | 10 | simp1d 1138 | . . . 4 ⊢ (𝜑 → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) |
12 | f1ofun 6611 | . . . 4 ⊢ (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) → Fun 𝐹) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
14 | 13 | adantr 483 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → Fun 𝐹) |
15 | ssun1 4147 | . . . 4 ⊢ 𝐺 ⊆ (𝐺 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) | |
16 | 15, 8 | sseqtrri 4003 | . . 3 ⊢ 𝐺 ⊆ 𝐹 |
17 | 16 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → 𝐺 ⊆ 𝐹) |
18 | f1odm 6613 | . . . . 5 ⊢ (𝐺:𝐾–1-1-onto→𝐾 → dom 𝐺 = 𝐾) | |
19 | 9, 18 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐺 = 𝐾) |
20 | 19 | eleq2d 2898 | . . 3 ⊢ (𝜑 → (𝑋 ∈ dom 𝐺 ↔ 𝑋 ∈ 𝐾)) |
21 | 20 | biimpar 480 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ dom 𝐺) |
22 | funssfv 6685 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝑋 ∈ dom 𝐺) → (𝐹‘𝑋) = (𝐺‘𝑋)) | |
23 | 14, 17, 21, 22 | syl3anc 1367 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝐺‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 ≠ wne 3016 ∀wral 3138 Vcvv 3494 ∖ cdif 3932 ∪ cun 3933 ⊆ wss 3935 {csn 4560 {cpr 4562 〈cop 4566 ↦ cmpt 5138 dom cdm 5549 Fun wfun 6343 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 Fincfn 8503 1c1 10532 + caddc 10534 ℕcn 11632 2c2 11686 ℕ0cn0 11891 ...cfz 12886 ♯chash 13684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-hash 13685 |
This theorem is referenced by: subfacp1lem3 32424 subfacp1lem4 32425 subfacp1lem5 32426 |
Copyright terms: Public domain | W3C validator |