| 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 35191. 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 35185 | . . . . 5 ⊢ (𝜑 → (𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1)) ∧ (𝐹‘1) = 𝑀 ∧ (𝐹‘𝑀) = 1)) |
| 11 | 10 | simp1d 1143 | . . . 4 ⊢ (𝜑 → 𝐹:(1...(𝑁 + 1))–1-1-onto→(1...(𝑁 + 1))) |
| 12 | f1ofun 6850 | . . . 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 4178 | . . . 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 6852 | . . . . 5 ⊢ (𝐺:𝐾–1-1-onto→𝐾 → dom 𝐺 = 𝐾) | |
| 19 | 9, 18 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐺 = 𝐾) |
| 20 | 19 | eleq2d 2827 | . . 3 ⊢ (𝜑 → (𝑋 ∈ dom 𝐺 ↔ 𝑋 ∈ 𝐾)) |
| 21 | 20 | biimpar 477 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → 𝑋 ∈ dom 𝐺) |
| 22 | funssfv 6927 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝑋 ∈ dom 𝐺) → (𝐹‘𝑋) = (𝐺‘𝑋)) | |
| 23 | 14, 17, 21, 22 | syl3anc 1373 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐾) → (𝐹‘𝑋) = (𝐺‘𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 {csn 4626 {cpr 4628 〈cop 4632 ↦ cmpt 5225 dom cdm 5685 Fun wfun 6555 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 1c1 11156 + caddc 11158 ℕcn 12266 2c2 12321 ℕ0cn0 12526 ...cfz 13547 ♯chash 14369 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 |
| This theorem is referenced by: subfacp1lem3 35187 subfacp1lem4 35188 subfacp1lem5 35189 |
| Copyright terms: Public domain | W3C validator |