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Theorem mapsnf1o 7991
Description: A bijection between a set and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypothesis
Ref Expression
ixpsnf1o.f 𝐹 = (𝑥𝐴 ↦ ({𝐼} × {𝑥}))
Assertion
Ref Expression
mapsnf1o ((𝐴𝑉𝐼𝑊) → 𝐹:𝐴1-1-onto→(𝐴𝑚 {𝐼}))
Distinct variable groups:   𝑥,𝐼   𝑥,𝐴   𝑥,𝑉   𝑥,𝑊
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem mapsnf1o
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ixpsnf1o.f . . . 4 𝐹 = (𝑥𝐴 ↦ ({𝐼} × {𝑥}))
21ixpsnf1o 7990 . . 3 (𝐼𝑊𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴)
32adantl 481 . 2 ((𝐴𝑉𝐼𝑊) → 𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴)
4 snex 4938 . . . . 5 {𝐼} ∈ V
5 ixpconstg 7959 . . . . . 6 (({𝐼} ∈ V ∧ 𝐴𝑉) → X𝑦 ∈ {𝐼}𝐴 = (𝐴𝑚 {𝐼}))
65eqcomd 2657 . . . . 5 (({𝐼} ∈ V ∧ 𝐴𝑉) → (𝐴𝑚 {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
74, 6mpan 706 . . . 4 (𝐴𝑉 → (𝐴𝑚 {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
87adantr 480 . . 3 ((𝐴𝑉𝐼𝑊) → (𝐴𝑚 {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
9 f1oeq3 6167 . . 3 ((𝐴𝑚 {𝐼}) = X𝑦 ∈ {𝐼}𝐴 → (𝐹:𝐴1-1-onto→(𝐴𝑚 {𝐼}) ↔ 𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴))
108, 9syl 17 . 2 ((𝐴𝑉𝐼𝑊) → (𝐹:𝐴1-1-onto→(𝐴𝑚 {𝐼}) ↔ 𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴))
113, 10mpbird 247 1 ((𝐴𝑉𝐼𝑊) → 𝐹:𝐴1-1-onto→(𝐴𝑚 {𝐼}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210  cmpt 4762   × cxp 5141  1-1-ontowf1o 5925  (class class class)co 6690  𝑚 cmap 7899  Xcixp 7950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-ixp 7951
This theorem is referenced by:  pwssnf1o  16205  mat1f1o  20332
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