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Mirrors > Home > MPE Home > Th. List > Mathboxes > subfacp1lem2b | Structured version Visualization version GIF version |
Description: Lemma for subfacp1 33671. 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 33665 | . . . . 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 6784 | . . . 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 481 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → Fun 𝐹) |
15 | ssun1 4131 | . . . 4 ⊢ 𝐺 ⊆ (𝐺 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) | |
16 | 15, 8 | sseqtrri 3980 | . . 3 ⊢ 𝐺 ⊆ 𝐹 |
17 | 16 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → 𝐺 ⊆ 𝐹) |
18 | f1odm 6786 | . . . . 5 ⊢ (𝐺:𝐾–1-1-onto→𝐾 → dom 𝐺 = 𝐾) | |
19 | 9, 18 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐺 = 𝐾) |
20 | 19 | eleq2d 2823 | . . 3 ⊢ (𝜑 → (𝑋 ∈ dom 𝐺 ↔ 𝑋 ∈ 𝐾)) |
21 | 20 | biimpar 478 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ dom 𝐺) |
22 | funssfv 6861 | . 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 396 = wceq 1541 ∈ wcel 2106 {cab 2713 ≠ wne 2942 ∀wral 3063 Vcvv 3444 ∖ cdif 3906 ∪ cun 3907 ⊆ wss 3909 {csn 4585 {cpr 4587 〈cop 4591 ↦ cmpt 5187 dom cdm 5632 Fun wfun 6488 –1-1-onto→wf1o 6493 ‘cfv 6494 (class class class)co 7354 Fincfn 8880 1c1 11049 + caddc 11051 ℕcn 12150 2c2 12205 ℕ0cn0 12410 ...cfz 13421 ♯chash 14227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-oadd 8413 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-dju 9834 df-card 9872 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-n0 12411 df-z 12497 df-uz 12761 df-fz 13422 df-hash 14228 |
This theorem is referenced by: subfacp1lem3 33667 subfacp1lem4 33668 subfacp1lem5 33669 |
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