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Mirrors > Home > MPE Home > Th. List > Mathboxes > subfacp1lem2b | Structured version Visualization version GIF version |
Description: Lemma for subfacp1 35171. 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 35165 | . . . . 5 ⊢ (𝜑 → (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘1) = 𝑀 ∧ (𝐹‘𝑀) = 1)) |
11 | 10 | simp1d 1141 | . . . 4 ⊢ (𝜑 → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) |
12 | f1ofun 6851 | . . . 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 4188 | . . . 4 ⊢ 𝐺 ⊆ (𝐺 ∪ {〈1, 𝑀〉, 〈𝑀, 1〉}) | |
16 | 15, 8 | sseqtrri 4033 | . . 3 ⊢ 𝐺 ⊆ 𝐹 |
17 | 16 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → 𝐺 ⊆ 𝐹) |
18 | f1odm 6853 | . . . . 5 ⊢ (𝐺:𝐾–1-1-onto→𝐾 → dom 𝐺 = 𝐾) | |
19 | 9, 18 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐺 = 𝐾) |
20 | 19 | eleq2d 2825 | . . 3 ⊢ (𝜑 → (𝑋 ∈ dom 𝐺 ↔ 𝑋 ∈ 𝐾)) |
21 | 20 | biimpar 477 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ dom 𝐺) |
22 | funssfv 6928 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝑋 ∈ dom 𝐺) → (𝐹‘𝑋) = (𝐺‘𝑋)) | |
23 | 14, 17, 21, 22 | syl3anc 1370 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝐺‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {cab 2712 ≠ wne 2938 ∀wral 3059 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ⊆ wss 3963 {csn 4631 {cpr 4633 〈cop 4637 ↦ cmpt 5231 dom cdm 5689 Fun wfun 6557 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 1c1 11154 + caddc 11156 ℕcn 12264 2c2 12319 ℕ0cn0 12524 ...cfz 13544 ♯chash 14366 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 |
This theorem is referenced by: subfacp1lem3 35167 subfacp1lem4 35168 subfacp1lem5 35169 |
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