Theorem mapsnf1o2 6865

Description: Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Hypotheses

Ref	Expression
mapsncnv.s	⊢ 𝑆 = {𝑋}
mapsncnv.b	⊢ 𝐵 ∈ V
mapsncnv.x	⊢ 𝑋 ∈ V
mapsncnv.f	⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋))

Assertion

Ref	Expression
mapsnf1o2	⊢ 𝐹:(𝐵 ↑𝑚 𝑆)–1-1-onto→𝐵

Distinct variable groups:   𝑥,𝐵   𝑥,𝑆

Allowed substitution hints:   𝐹(𝑥)   𝑋(𝑥)

Proof of Theorem mapsnf1o2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2805	. . . 4 ⊢ 𝑥 ∈ V
2		mapsncnv.x	. . . 4 ⊢ 𝑋 ∈ V
3	1, 2	fvex 5659	. . 3 ⊢ (𝑥‘𝑋) ∈ V
4		mapsncnv.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋))
5	3, 4	fnmpti 5461	. 2 ⊢ 𝐹 Fn (𝐵 ↑𝑚 𝑆)
6		mapsncnv.s	. . . . 5 ⊢ 𝑆 = {𝑋}
7	2	snex 4275	. . . . 5 ⊢ {𝑋} ∈ V
8	6, 7	eqeltri 2304	. . . 4 ⊢ 𝑆 ∈ V
9		vex 2805	. . . . 5 ⊢ 𝑦 ∈ V
10	9	snex 4275	. . . 4 ⊢ {𝑦} ∈ V
11	8, 10	xpex 4842	. . 3 ⊢ (𝑆 × {𝑦}) ∈ V
12		mapsncnv.b	. . . 4 ⊢ 𝐵 ∈ V
13	6, 12, 2, 4	mapsncnv 6864	. . 3 ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
14	11, 13	fnmpti 5461	. 2 ⊢ ◡𝐹 Fn 𝐵
15		dff1o4 5591	. 2 ⊢ (𝐹:(𝐵 ↑𝑚 𝑆)–1-1-onto→𝐵 ↔ (𝐹 Fn (𝐵 ↑𝑚 𝑆) ∧ ◡𝐹 Fn 𝐵))
16	5, 14, 15	mpbir2an 950	1 ⊢ 𝐹:(𝐵 ↑𝑚 𝑆)–1-1-onto→𝐵

Syntax hints:   = wceq 1397   ∈ wcel 2202  Vcvv 2802  {csn 3669   ↦ cmpt 4150   × cxp 4723  ^◡ccnv 4724   Fn wfn 5321  –1-1-onto→wf1o 5325  ‘cfv 5326  (class class class)co 6018   ↑_𝑚 cmap 6817

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-map 6819

This theorem is referenced by:  mapsnf1o3  6866