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Theorem mapsnf1o 7835
 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 7834 . . 3 (𝐼𝑊𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴)
32adantl 481 . 2 ((𝐴𝑉𝐼𝑊) → 𝐹:𝐴1-1-ontoX𝑦 ∈ {𝐼}𝐴)
4 snex 4835 . . . . 5 {𝐼} ∈ V
5 ixpconstg 7803 . . . . . 6 (({𝐼} ∈ V ∧ 𝐴𝑉) → X𝑦 ∈ {𝐼}𝐴 = (𝐴𝑚 {𝐼}))
65eqcomd 2616 . . . . 5 (({𝐼} ∈ V ∧ 𝐴𝑉) → (𝐴𝑚 {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
74, 6mpan 702 . . . 4 (𝐴𝑉 → (𝐴𝑚 {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
87adantr 480 . . 3 ((𝐴𝑉𝐼𝑊) → (𝐴𝑚 {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
9 f1oeq3 6042 . . 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 246 1 ((𝐴𝑉𝐼𝑊) → 𝐹:𝐴1-1-onto→(𝐴𝑚 {𝐼}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125   ↦ cmpt 4643   × cxp 5036  –1-1-onto→wf1o 5803  (class class class)co 6549   ↑𝑚 cmap 7744  Xcixp 7794 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-ixp 7795 This theorem is referenced by:  pwssnf1o  15981  mat1f1o  20103
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