mapsnf1o - Metamath Proof Explorer

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Theorem mapsnf1o 8992

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⟶1-1onto(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 8991	. . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐹:𝐴⟶1-1ontoX𝑦 ∈ {𝐼}𝐴)
3	2	adantl 487	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐴⟶1-1ontoX𝑦 ∈ {𝐼}𝐴)
4		snex 5401	. . . . 5 ⊢ {𝐼} ∈ V
5		ixpconstg 8959	. . . . . 6 ⊢ (({𝐼} ∈ V ∧ 𝐴 ∈ 𝑉) → X𝑦 ∈ {𝐼}𝐴 = (𝐴 ↑_m {𝐼}))
6	5	eqcomd 2769	. . . . 5 ⊢ (({𝐼} ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐴 ↑_m {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
7	4, 6	mpan 703	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑_m {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
8	7	adantr 486	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝐴 ↑_m {𝐼}) = X𝑦 ∈ {𝐼}𝐴)
9	8	f1oeq3d 6840	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝐹:𝐴⟶1-1onto(𝐴 ↑_m {𝐼}) ↔ 𝐹:𝐴⟶1-1ontoX𝑦 ∈ {𝐼}𝐴))
10	3, 9	mpbird 260	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐴⟶1-1onto(𝐴 ↑_m {𝐼}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 Vcvv 3454 {csn 4588 ↦ cmpt 5189 × cxp 5653 ⟶1-1ontowf1o 6554 (class class class)co 7440 ↑_m cmap 8872 Xcixp 8950

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-ov 7443 df-oprab 7444 df-mpo 7445 df-map 8874 df-ixp 8951

This theorem is used by: pwssnf1o 17696 mat1f1o 22820

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