Theorem mapsnf1o 8685

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 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥}))

Assertion

Ref	Expression
mapsnf1o	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐴–1-1-onto→(𝐴 ↑m {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.

Step	Hyp	Ref	Expression
1		ixpsnf1o.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥}))
2	1	ixpsnf1o 8684	. . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴)
3	2	adantl 481	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴)
4		snex 5349	. . . . 5 ⊢ {𝐼} ∈ V
5		ixpconstg 8652	. . . . . 6 ⊢ (({𝐼} ∈ V ∧ 𝐴 ∈ 𝑉) → X𝑦 ∈ {𝐼}𝐴 = (𝐴 ↑m {𝐼}))
6	5	eqcomd 2744	. . . . 5 ⊢ (({𝐼} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ↑m {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
7	4, 6	mpan 686	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
8	7	adantr 480	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝐴 ↑m {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
9	8	f1oeq3d 6697	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝐹:𝐴–1-1-onto→(𝐴 ↑m {𝐼}) ↔ 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴))
10	3, 9	mpbird 256	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐴–1-1-onto→(𝐴 ↑m {𝐼}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  {csn 4558   ↦ cmpt 5153   × cxp 5578  –1-1-onto→wf1o 6417  (class class class)co 7255   ↑_m cmap 8573  Xcixp 8643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-ixp 8644

This theorem is referenced by:  pwssnf1o  17126  mat1f1o  21535