Theorem wrd2f1tovbij 14913

Description: There is a bijection between words of length two with a fixed first symbol contained in a pair and the symbols contained in a pair together with the fixed symbol. (Contributed by Alexander van der Vekens, 28-Jul-2018.)

Assertion

Ref	Expression
wrd2f1tovbij	⊢ ((𝑉 ∈ 𝑌 ∧ 𝑃 ∈ 𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋})

Distinct variable groups:   𝑃,𝑓,𝑛,𝑤   𝑓,𝑉,𝑛,𝑤   𝑓,𝑋,𝑛,𝑤

Allowed substitution hints:   𝑌(𝑤,𝑓,𝑛)

Proof of Theorem wrd2f1tovbij

Dummy variables 𝑝 𝑡 𝑢 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wrdexg 14476	. . . 4 ⊢ (𝑉 ∈ 𝑌 → Word 𝑉 ∈ V)
2	1	adantr 481	. . 3 ⊢ ((𝑉 ∈ 𝑌 ∧ 𝑃 ∈ 𝑉) → Word 𝑉 ∈ V)
3		rabexg 5331	. . 3 ⊢ (Word 𝑉 ∈ V → {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ∈ V)
4		mptexg 7225	. . 3 ⊢ ({𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ∈ V → (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ↦ (𝑥‘1)) ∈ V)
5	2, 3, 4	3syl 18	. 2 ⊢ ((𝑉 ∈ 𝑌 ∧ 𝑃 ∈ 𝑉) → (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ↦ (𝑥‘1)) ∈ V)
6		fveqeq2 6900	. . . . . 6 ⊢ (𝑤 = 𝑢 → ((♯‘𝑤) = 2 ↔ (♯‘𝑢) = 2))
7		fveq1 6890	. . . . . . 7 ⊢ (𝑤 = 𝑢 → (𝑤‘0) = (𝑢‘0))
8	7	eqeq1d 2734	. . . . . 6 ⊢ (𝑤 = 𝑢 → ((𝑤‘0) = 𝑃 ↔ (𝑢‘0) = 𝑃))
9		fveq1 6890	. . . . . . . 8 ⊢ (𝑤 = 𝑢 → (𝑤‘1) = (𝑢‘1))
10	7, 9	preq12d 4745	. . . . . . 7 ⊢ (𝑤 = 𝑢 → {(𝑤‘0), (𝑤‘1)} = {(𝑢‘0), (𝑢‘1)})
11	10	eleq1d 2818	. . . . . 6 ⊢ (𝑤 = 𝑢 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋))
12	6, 8, 11	3anbi123d 1436	. . . . 5 ⊢ (𝑤 = 𝑢 → (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)))
13	12	cbvrabv 3442	. . . 4 ⊢ {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} = {𝑢 ∈ Word 𝑉 ∣ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)}
14		preq2 4738	. . . . . 6 ⊢ (𝑛 = 𝑝 → {𝑃, 𝑛} = {𝑃, 𝑝})
15	14	eleq1d 2818	. . . . 5 ⊢ (𝑛 = 𝑝 → ({𝑃, 𝑛} ∈ 𝑋 ↔ {𝑃, 𝑝} ∈ 𝑋))
16	15	cbvrabv 3442	. . . 4 ⊢ {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} = {𝑝 ∈ 𝑉 ∣ {𝑃, 𝑝} ∈ 𝑋}
17		fveqeq2 6900	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((♯‘𝑡) = 2 ↔ (♯‘𝑤) = 2))
18		fveq1 6890	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (𝑡‘0) = (𝑤‘0))
19	18	eqeq1d 2734	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((𝑡‘0) = 𝑃 ↔ (𝑤‘0) = 𝑃))
20		fveq1 6890	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → (𝑡‘1) = (𝑤‘1))
21	18, 20	preq12d 4745	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → {(𝑡‘0), (𝑡‘1)} = {(𝑤‘0), (𝑤‘1)})
22	21	eleq1d 2818	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ({(𝑡‘0), (𝑡‘1)} ∈ 𝑋 ↔ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋))
23	17, 19, 22	3anbi123d 1436	. . . . . 6 ⊢ (𝑡 = 𝑤 → (((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋) ↔ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)))
24	23	cbvrabv 3442	. . . . 5 ⊢ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}
25	24	mpteq1i 5244	. . . 4 ⊢ (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ↦ (𝑥‘1)) = (𝑥 ∈ {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)} ↦ (𝑥‘1))
26	13, 16, 25	wwlktovf1o 14912	. . 3 ⊢ (𝑃 ∈ 𝑉 → (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ↦ (𝑥‘1)):{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋})
27	26	adantl 482	. 2 ⊢ ((𝑉 ∈ 𝑌 ∧ 𝑃 ∈ 𝑉) → (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ↦ (𝑥‘1)):{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋})
28		f1oeq1 6821	. 2 ⊢ (𝑓 = (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ↦ (𝑥‘1)) → (𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} ↔ (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ↦ (𝑥‘1)):{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}))
29	5, 27, 28	spcedv 3588	1 ⊢ ((𝑉 ∈ 𝑌 ∧ 𝑃 ∈ 𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋})

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {crab 3432  Vcvv 3474  {cpr 4630   ↦ cmpt 5231  –1-1-onto→wf1o 6542  ‘cfv 6543  0cc0 11112  1c1 11113  2c2 12269  ♯chash 14292  Word cword 14466

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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-dju 9898  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-n0 12475  df-z 12561  df-uz 12825  df-fz 13487  df-fzo 13630  df-hash 14293  df-word 14467

This theorem is referenced by:  rusgrnumwrdl2  28881