Theorem wrd2f1tovbij 14911

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 14474	. . . 4 ⊢ (𝑉 ∈ 𝑌 → Word 𝑉 ∈ V)
2	1	adantr 482	. . 3 ⊢ ((𝑉 ∈ 𝑌 ∧ 𝑃 ∈ 𝑉) → Word 𝑉 ∈ V)
3		rabexg 5332	. . 3 ⊢ (Word 𝑉 ∈ V → {𝑡 ∈ Word 𝑉 ∣ ((♯‘𝑡) = 2 ∧ (𝑡‘0) = 𝑃 ∧ {(𝑡‘0), (𝑡‘1)} ∈ 𝑋)} ∈ V)
4		mptexg 7223	. . 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 6901	. . . . . 6 ⊢ (𝑤 = 𝑢 → ((♯‘𝑤) = 2 ↔ (♯‘𝑢) = 2))
7		fveq1 6891	. . . . . . 7 ⊢ (𝑤 = 𝑢 → (𝑤‘0) = (𝑢‘0))
8	7	eqeq1d 2735	. . . . . 6 ⊢ (𝑤 = 𝑢 → ((𝑤‘0) = 𝑃 ↔ (𝑢‘0) = 𝑃))
9		fveq1 6891	. . . . . . . 8 ⊢ (𝑤 = 𝑢 → (𝑤‘1) = (𝑢‘1))
10	7, 9	preq12d 4746	. . . . . . 7 ⊢ (𝑤 = 𝑢 → {(𝑤‘0), (𝑤‘1)} = {(𝑢‘0), (𝑢‘1)})
11	10	eleq1d 2819	. . . . . 6 ⊢ (𝑤 = 𝑢 → ({(𝑤‘0), (𝑤‘1)} ∈ 𝑋 ↔ {(𝑢‘0), (𝑢‘1)} ∈ 𝑋))
12	6, 8, 11	3anbi123d 1437	. . . . 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 3443	. . . 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 4739	. . . . . 6 ⊢ (𝑛 = 𝑝 → {𝑃, 𝑛} = {𝑃, 𝑝})
15	14	eleq1d 2819	. . . . 5 ⊢ (𝑛 = 𝑝 → ({𝑃, 𝑛} ∈ 𝑋 ↔ {𝑃, 𝑝} ∈ 𝑋))
16	15	cbvrabv 3443	. . . 4 ⊢ {𝑛 ∈ 𝑉 ∣ {𝑃, 𝑛} ∈ 𝑋} = {𝑝 ∈ 𝑉 ∣ {𝑃, 𝑝} ∈ 𝑋}
17		fveqeq2 6901	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((♯‘𝑡) = 2 ↔ (♯‘𝑤) = 2))
18		fveq1 6891	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → (𝑡‘0) = (𝑤‘0))
19	18	eqeq1d 2735	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ((𝑡‘0) = 𝑃 ↔ (𝑤‘0) = 𝑃))
20		fveq1 6891	. . . . . . . . 9 ⊢ (𝑡 = 𝑤 → (𝑡‘1) = (𝑤‘1))
21	18, 20	preq12d 4746	. . . . . . . 8 ⊢ (𝑡 = 𝑤 → {(𝑡‘0), (𝑡‘1)} = {(𝑤‘0), (𝑤‘1)})
22	21	eleq1d 2819	. . . . . . 7 ⊢ (𝑡 = 𝑤 → ({(𝑡‘0), (𝑡‘1)} ∈ 𝑋 ↔ {(𝑤‘0), (𝑤‘1)} ∈ 𝑋))
23	17, 19, 22	3anbi123d 1437	. . . . . 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 3443	. . . . 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 5245	. . . 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 14910	. . 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 483	. 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 6822	. 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 3589	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 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {crab 3433  Vcvv 3475  {cpr 4631   ↦ cmpt 5232  –1-1-onto→wf1o 6543  ‘cfv 6544  0cc0 11110  1c1 11111  2c2 12267  ♯chash 14290  Word cword 14464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-dju 9896  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465

This theorem is referenced by:  rusgrnumwrdl2  28843