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Theorem wrd2f1tovbij 14316
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 ((𝑉𝑌𝑃𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛𝑉 ∣ {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.
StepHypRef Expression
1 wrdexg 13863 . . . 4 (𝑉𝑌 → Word 𝑉 ∈ V)
21adantr 483 . . 3 ((𝑉𝑌𝑃𝑉) → Word 𝑉 ∈ V)
3 rabexg 5225 . . 3 (Word 𝑉 ∈ V → {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ∈ V)
4 mptexg 6976 . . 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)
52, 3, 43syl 18 . 2 ((𝑉𝑌𝑃𝑉) → (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ↦ (𝑥‘1)) ∈ V)
6 fveqeq2 6672 . . . . . 6 (𝑤 = 𝑢 → ((♯‘𝑤) = 2 ↔ (♯‘𝑢) = 2))
7 fveq1 6662 . . . . . . 7 (𝑤 = 𝑢 → (𝑤‘0) = (𝑢‘0))
87eqeq1d 2821 . . . . . 6 (𝑤 = 𝑢 → ((𝑤‘0) = 𝑃 ↔ (𝑢‘0) = 𝑃))
9 fveq1 6662 . . . . . . . 8 (𝑤 = 𝑢 → (𝑤‘1) = (𝑢‘1))
107, 9preq12d 4669 . . . . . . 7 (𝑤 = 𝑢 → {(𝑤‘0), (𝑤‘1)} = {(𝑢‘0), (𝑢‘1)})
1110eleq1d 2895 . . . . . 6 (𝑤 = 𝑢 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋))
126, 8, 113anbi123d 1429 . . . . 5 (𝑤 = 𝑢 → (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)))
1312cbvrabv 3490 . . . 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 4662 . . . . . 6 (𝑛 = 𝑝 → {𝑃, 𝑛} = {𝑃, 𝑝})
1514eleq1d 2895 . . . . 5 (𝑛 = 𝑝 → ({𝑃, 𝑛} ∈ 𝑋 ↔ {𝑃, 𝑝} ∈ 𝑋))
1615cbvrabv 3490 . . . 4 {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} = {𝑝𝑉 ∣ {𝑃, 𝑝} ∈ 𝑋}
17 fveqeq2 6672 . . . . . . 7 (𝑡 = 𝑤 → ((♯‘𝑡) = 2 ↔ (♯‘𝑤) = 2))
18 fveq1 6662 . . . . . . . 8 (𝑡 = 𝑤 → (𝑡‘0) = (𝑤‘0))
1918eqeq1d 2821 . . . . . . 7 (𝑡 = 𝑤 → ((𝑡‘0) = 𝑃 ↔ (𝑤‘0) = 𝑃))
20 fveq1 6662 . . . . . . . . 9 (𝑡 = 𝑤 → (𝑡‘1) = (𝑤‘1))
2118, 20preq12d 4669 . . . . . . . 8 (𝑡 = 𝑤 → {(𝑡‘0), (𝑡‘1)} = {(𝑤‘0), (𝑤‘1)})
2221eleq1d 2895 . . . . . . 7 (𝑡 = 𝑤 → ({(𝑡‘0), (𝑡‘1)} ∈ 𝑋 ↔ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋))
2317, 19, 223anbi123d 1429 . . . . . 6 (𝑡 = 𝑤 → (((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋) ↔ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)))
2423cbvrabv 3490 . . . . 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)}
2524mpteq1i 5147 . . . 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))
2613, 16, 25wwlktovf1o 14315 . . 3 (𝑃𝑉 → (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→{𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋})
2726adantl 484 . 2 ((𝑉𝑌𝑃𝑉) → (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→{𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋})
28 f1oeq1 6597 . 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→{𝑛𝑉 ∣ {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→{𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}))
295, 27, 28spcedv 3597 1 ((𝑉𝑌𝑃𝑉) → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1081   = wceq 1530  wex 1773  wcel 2107  {crab 3140  Vcvv 3493  {cpr 4561  cmpt 5137  1-1-ontowf1o 6347  cfv 6348  0cc0 10529  1c1 10530  2c2 11684  chash 13682  Word cword 13853
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026  df-hash 13683  df-word 13854
This theorem is referenced by:  rusgrnumwrdl2  27360
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