Theorem mapsnf1o3 6861

Description: Explicit bijection in the reverse of mapsnf1o2 6860. (Contributed by Stefan O'Rear, 24-Mar-2015.)

Hypotheses

Ref	Expression
mapsncnv.s	⊢ 𝑆 = {𝑋}
mapsncnv.b	⊢ 𝐵 ∈ V
mapsncnv.x	⊢ 𝑋 ∈ V
mapsnf1o3.f	⊢ 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))

Assertion

Ref	Expression
mapsnf1o3	⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑𝑚 𝑆)

Distinct variable groups:   𝑦,𝐵   𝑦,𝑆   𝑦,𝑋

Allowed substitution hint:   𝐹(𝑦)

Proof of Theorem mapsnf1o3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapsncnv.s	. . . 4 ⊢ 𝑆 = {𝑋}
2		mapsncnv.b	. . . 4 ⊢ 𝐵 ∈ V
3		mapsncnv.x	. . . 4 ⊢ 𝑋 ∈ V
4		eqid 2229	. . . 4 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)) = (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋))
5	1, 2, 3, 4	mapsnf1o2 6860	. . 3 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑𝑚 𝑆)–1-1-onto→𝐵
6		f1ocnv 5593	. . 3 ⊢ ((𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑𝑚 𝑆)–1-1-onto→𝐵 → ◡(𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)):𝐵–1-1-onto→(𝐵 ↑𝑚 𝑆))
7	5, 6	ax-mp 5	. 2 ⊢ ◡(𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)):𝐵–1-1-onto→(𝐵 ↑𝑚 𝑆)
8		mapsnf1o3.f	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
9	1, 2, 3, 4	mapsncnv 6859	. . . 4 ⊢ ◡(𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)) = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
10	8, 9	eqtr4i 2253	. . 3 ⊢ 𝐹 = ◡(𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋))
11		f1oeq1 5568	. . 3 ⊢ (𝐹 = ◡(𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)) → (𝐹:𝐵–1-1-onto→(𝐵 ↑𝑚 𝑆) ↔ ◡(𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)):𝐵–1-1-onto→(𝐵 ↑𝑚 𝑆)))
12	10, 11	ax-mp 5	. 2 ⊢ (𝐹:𝐵–1-1-onto→(𝐵 ↑𝑚 𝑆) ↔ ◡(𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)):𝐵–1-1-onto→(𝐵 ↑𝑚 𝑆))
13	7, 12	mpbir 146	1 ⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑𝑚 𝑆)

Syntax hints:   ↔ wb 105   = wceq 1395   ∈ wcel 2200  Vcvv 2800  {csn 3667   ↦ cmpt 4148   × cxp 4721  ^◡ccnv 4722  –1-1-onto→wf1o 5323  ‘cfv 5324  (class class class)co 6013   ↑_𝑚 cmap 6812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-map 6814

This theorem is referenced by: (None)