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Theorem mapsnf1o 8550
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 8549 . . 3 (𝐼𝑊𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴)
32adantl 485 . 2 ((𝐴𝑉𝐼𝑊) → 𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴)
4 snex 5299 . . . . 5 {𝐼} ∈ V
5 ixpconstg 8517 . . . . . 6 (({𝐼} ∈ V ∧ 𝐴𝑉) → X𝑦 ∈ {𝐼}𝐴 = (𝐴m {𝐼}))
65eqcomd 2744 . . . . 5 (({𝐼} ∈ V ∧ 𝐴𝑉) → (𝐴m {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
74, 6mpan 690 . . . 4 (𝐴𝑉 → (𝐴m {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
87adantr 484 . . 3 ((𝐴𝑉𝐼𝑊) → (𝐴m {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
98f1oeq3d 6616 . 2 ((𝐴𝑉𝐼𝑊) → (𝐹:𝐴1-1-onto→(𝐴m {𝐼}) ↔ 𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴))
103, 9mpbird 260 1 ((𝐴𝑉𝐼𝑊) → 𝐹:𝐴1-1-onto→(𝐴m {𝐼}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2113  Vcvv 3398  {csn 4517  cmpt 5111   × cxp 5524  1-1-ontowf1o 6339  (class class class)co 7171  m cmap 8438  Xcixp 8508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7480
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3683  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7174  df-oprab 7175  df-mpo 7176  df-map 8440  df-ixp 8509
This theorem is referenced by:  pwssnf1o  16875  mat1f1o  21230
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