Theorem mapsnf1o3 6521

Description: Explicit bijection in the reverse of mapsnf1o2 6520. (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 2100	. . . 4 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)) = (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋))
5	1, 2, 3, 4	mapsnf1o2 6520	. . 3 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑𝑚 𝑆)–1-1-onto→𝐵
6		f1ocnv 5314	. . 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 7	. 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 6519	. . . 4 ⊢ ◡(𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)) = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
10	8, 9	eqtr4i 2123	. . 3 ⊢ 𝐹 = ◡(𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋))
11		f1oeq1 5292	. . 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 7	. 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 145	1 ⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑𝑚 𝑆)

Syntax hints:   ↔ wb 104   = wceq 1299   ∈ wcel 1448  Vcvv 2641  {csn 3474   ↦ cmpt 3929   × cxp 4475  ^◡ccnv 4476  –1-1-onto→wf1o 5058  ‘cfv 5059  (class class class)co 5706   ↑_𝑚 cmap 6472

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-map 6474

This theorem is referenced by: (None)