Theorem wrd2f1tovbij 14908

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 14471	. . . 4 ⊢ (𝑉 ∈ 𝑌 → Word 𝑉 ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝑉 ∈ 𝑌 ∧ 𝑃 ∈ 𝑉) → Word 𝑉 ∈ V)
3		rabexg 5321	. . 3 ⊢ (Word 𝑉 ∈ V → {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ∈ V)
4		mptexg 7214	. . 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 6890	. . . . . 6 ⊢ (𝑤 = 𝑢 → ((♯‘𝑤) = 2 ↔ (♯‘𝑢) = 2))
7		fveq1 6880	. . . . . . 7 ⊢ (𝑤 = 𝑢 → (𝑤‘0) = (𝑢‘0))
8	7	eqeq1d 2726	. . . . . 6 ⊢ (𝑤 = 𝑢 → ((𝑤‘0) = 𝑃 ↔ (𝑢‘0) = 𝑃))
9		fveq1 6880	. . . . . . . 8 ⊢ (𝑤 = 𝑢 → (𝑤‘1) = (𝑢‘1))
10	7, 9	preq12d 4737	. . . . . . 7 ⊢ (𝑤 = 𝑢 → {(𝑤‘0), (𝑤‘1)} = {(𝑢‘0), (𝑢‘1)})
11	10	eleq1d 2810	. . . . . 6 ⊢ (𝑤 = 𝑢 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋))
12	6, 8, 11	3anbi123d 1432	. . . . 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 3434	. . . 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 4730	. . . . . 6 ⊢ (𝑛 = 𝑝 → {𝑃, 𝑛} = {𝑃, 𝑝})
15	14	eleq1d 2810	. . . . 5 ⊢ (𝑛 = 𝑝 → ({𝑃, 𝑛} ∈ 𝑋 ↔ {𝑃, 𝑝} ∈ 𝑋))
16	15	cbvrabv 3434	. . . 4 ⊢ {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} = {𝑝 ∈ 𝑉 ∣ {𝑃, 𝑝} ∈ 𝑋}
17		fveqeq2 6890	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((♯‘𝑡) = 2 ↔ (♯‘𝑤) = 2))
18		fveq1 6880	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (𝑡‘0) = (𝑤‘0))
19	18	eqeq1d 2726	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((𝑡‘0) = 𝑃 ↔ (𝑤‘0) = 𝑃))
20		fveq1 6880	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → (𝑡‘1) = (𝑤‘1))
21	18, 20	preq12d 4737	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → {(𝑡‘0), (𝑡‘1)} = {(𝑤‘0), (𝑤‘1)})
22	21	eleq1d 2810	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ({(𝑡‘0), (𝑡‘1)} ∈ 𝑋 ↔ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋))
23	17, 19, 22	3anbi123d 1432	. . . . . 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 3434	. . . . 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 5234	. . . 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 14907	. . 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 481	. 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 6811	. 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 3580	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 395   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {crab 3424  Vcvv 3466  {cpr 4622   ↦ cmpt 5221  –1-1-onto→wf1o 6532  ‘cfv 6533  0cc0 11106  1c1 11107  2c2 12264  ♯chash 14287  Word cword 14461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-dju 9892  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-fzo 13625  df-hash 14288  df-word 14462

This theorem is referenced by:  rusgrnumwrdl2  29312