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Theorem wrd2f1tovbij 13904
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 13501 . . . 4 (𝑉𝑌 → Word 𝑉 ∈ V)
21adantr 472 . . 3 ((𝑉𝑌𝑃𝑉) → Word 𝑉 ∈ V)
3 rabexg 4963 . . 3 (Word 𝑉 ∈ V → {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ∈ V)
4 mptexg 6648 . . 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 fveq2 6352 . . . . . . 7 (𝑤 = 𝑢 → (♯‘𝑤) = (♯‘𝑢))
76eqeq1d 2762 . . . . . 6 (𝑤 = 𝑢 → ((♯‘𝑤) = 2 ↔ (♯‘𝑢) = 2))
8 fveq1 6351 . . . . . . 7 (𝑤 = 𝑢 → (𝑤‘0) = (𝑢‘0))
98eqeq1d 2762 . . . . . 6 (𝑤 = 𝑢 → ((𝑤‘0) = 𝑃 ↔ (𝑢‘0) = 𝑃))
10 fveq1 6351 . . . . . . . 8 (𝑤 = 𝑢 → (𝑤‘1) = (𝑢‘1))
118, 10preq12d 4420 . . . . . . 7 (𝑤 = 𝑢 → {(𝑤‘0), (𝑤‘1)} = {(𝑢‘0), (𝑢‘1)})
1211eleq1d 2824 . . . . . 6 (𝑤 = 𝑢 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋))
137, 9, 123anbi123d 1548 . . . . 5 (𝑤 = 𝑢 → (((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋) ↔ ((♯‘𝑢) = 2 ∧ (𝑢‘0) = 𝑃 ∧ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋)))
1413cbvrabv 3339 . . . 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)}
15 preq2 4413 . . . . . 6 (𝑛 = 𝑝 → {𝑃, 𝑛} = {𝑃, 𝑝})
1615eleq1d 2824 . . . . 5 (𝑛 = 𝑝 → ({𝑃, 𝑛} ∈ 𝑋 ↔ {𝑃, 𝑝} ∈ 𝑋))
1716cbvrabv 3339 . . . 4 {𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} = {𝑝𝑉 ∣ {𝑃, 𝑝} ∈ 𝑋}
18 fveq2 6352 . . . . . . . 8 (𝑡 = 𝑤 → (♯‘𝑡) = (♯‘𝑤))
1918eqeq1d 2762 . . . . . . 7 (𝑡 = 𝑤 → ((♯‘𝑡) = 2 ↔ (♯‘𝑤) = 2))
20 fveq1 6351 . . . . . . . 8 (𝑡 = 𝑤 → (𝑡‘0) = (𝑤‘0))
2120eqeq1d 2762 . . . . . . 7 (𝑡 = 𝑤 → ((𝑡‘0) = 𝑃 ↔ (𝑤‘0) = 𝑃))
22 fveq1 6351 . . . . . . . . 9 (𝑡 = 𝑤 → (𝑡‘1) = (𝑤‘1))
2320, 22preq12d 4420 . . . . . . . 8 (𝑡 = 𝑤 → {(𝑡‘0), (𝑡‘1)} = {(𝑤‘0), (𝑤‘1)})
2423eleq1d 2824 . . . . . . 7 (𝑡 = 𝑤 → ({(𝑡‘0), (𝑡‘1)} ∈ 𝑋 ↔ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋))
2519, 21, 243anbi123d 1548 . . . . . 6 (𝑡 = 𝑤 → (((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋) ↔ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)))
2625cbvrabv 3339 . . . . 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)}
27 mpteq1 4889 . . . . 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)} → (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)))
2826, 27ax-mp 5 . . . 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))
2914, 17, 28wwlktovf1o 13903 . . 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})
3029adantl 473 . 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})
31 f1oeq1 6288 . . 3 (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}))
3231spcegv 3434 . 2 ((𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ↦ (𝑥‘1)) ∈ 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→{𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} → ∃𝑓 𝑓:{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋)}–1-1-onto→{𝑛𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋}))
335, 30, 32sylc 65 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 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  {crab 3054  Vcvv 3340  {cpr 4323  cmpt 4881  1-1-ontowf1o 6048  cfv 6049  0cc0 10128  1c1 10129  2c2 11262  chash 13311  Word cword 13477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-card 8955  df-cda 9182  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-fzo 12660  df-hash 13312  df-word 13485
This theorem is referenced by:  rusgrnumwrdl2  26692
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