Theorem mapsnf1o 8881

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 8880	. . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴)
3	2	adantl 483	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴)
4		snex 5371	. . . . 5 ⊢ {𝐼} ∈ V
5		ixpconstg 8848	. . . . . 6 ⊢ (({𝐼} ∈ V ∧ 𝐴 ∈ 𝑉) → X𝑦 ∈ {𝐼}𝐴 = (𝐴 ↑m {𝐼}))
6	5	eqcomd 2747	. . . . 5 ⊢ (({𝐼} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ↑m {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
7	4, 6	mpan 697	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
8	7	adantr 482	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝐴 ↑m {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
9	8	f1oeq3d 6768	. 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 259	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐴–1-1-onto→(𝐴 ↑m {𝐼}))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  Vcvv 3433  {csn 4558   ↦ cmpt 5156   × cxp 5619  –1-1-onto→wf1o 6488  (class class class)co 7360   ↑_m cmap 8767  Xcixp 8839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8769  df-ixp 8840

This theorem is referenced by:  pwssnf1o  17457  mat1f1o  22465