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Mirrors > Home > MPE Home > Th. List > Mathboxes > subfacp1lem2b | Structured version Visualization version GIF version |
Description: Lemma for subfacp1 35154. 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 35148 | . . . . 5 ⊢ (𝜑 → (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘1) = 𝑀 ∧ (𝐹‘𝑀) = 1)) |
11 | 10 | simp1d 1142 | . . . 4 ⊢ (𝜑 → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) |
12 | f1ofun 6864 | . . . 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 480 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → Fun 𝐹) |
15 | ssun1 4201 | . . . 4 ⊢ 𝐺 ⊆ (𝐺 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) | |
16 | 15, 8 | sseqtrri 4046 | . . 3 ⊢ 𝐺 ⊆ 𝐹 |
17 | 16 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → 𝐺 ⊆ 𝐹) |
18 | f1odm 6866 | . . . . 5 ⊢ (𝐺:𝐾–1-1-onto→𝐾 → dom 𝐺 = 𝐾) | |
19 | 9, 18 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐺 = 𝐾) |
20 | 19 | eleq2d 2830 | . . 3 ⊢ (𝜑 → (𝑋 ∈ dom 𝐺 ↔ 𝑋 ∈ 𝐾)) |
21 | 20 | biimpar 477 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ dom 𝐺) |
22 | funssfv 6941 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝑋 ∈ dom 𝐺) → (𝐹‘𝑋) = (𝐺‘𝑋)) | |
23 | 14, 17, 21, 22 | syl3anc 1371 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝐺‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 {cab 2717 ≠ wne 2946 ∀wral 3067 Vcvv 3488 ∖ cdif 3973 ∪ cun 3974 ⊆ wss 3976 {csn 4648 {cpr 4650 〈cop 4654 ↦ cmpt 5249 dom cdm 5700 Fun wfun 6567 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 Fincfn 9003 1c1 11185 + caddc 11187 ℕcn 12293 2c2 12348 ℕ0cn0 12553 ...cfz 13567 ♯chash 14379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-hash 14380 |
This theorem is referenced by: subfacp1lem3 35150 subfacp1lem4 35151 subfacp1lem5 35152 |
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