Theorem mapsnf1o2 6983

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 2824	. . . 4 ⊢ 𝑥 ∈ V
2		mapsncnv.x	. . . 4 ⊢ 𝑋 ∈ V
3	1, 2	fvex 5718	. . 3 ⊢ (𝑥‘𝑋) ∈ V
4		mapsncnv.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋))
5	3, 4	fnmpti 5515	. 2 ⊢ 𝐹 Fn (𝐵 ↑𝑚 𝑆)
6		mapsncnv.s	. . . . 5 ⊢ 𝑆 = {𝑋}
7	2	snex 4324	. . . . 5 ⊢ {𝑋} ∈ V
8	6, 7	eqeltri 2311	. . . 4 ⊢ 𝑆 ∈ V
9		vex 2824	. . . . 5 ⊢ 𝑦 ∈ V
10	9	snex 4324	. . . 4 ⊢ {𝑦} ∈ V
11	8, 10	xpex 4893	. . 3 ⊢ (𝑆 × {𝑦}) ∈ V
12		mapsncnv.b	. . . 4 ⊢ 𝐵 ∈ V
13	6, 12, 2, 4	mapsncnv 6982	. . 3 ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
14	11, 13	fnmpti 5515	. 2 ⊢ ◡𝐹 Fn 𝐵
15		dff1o4 5650	. 2 ⊢ (𝐹:(𝐵 ↑𝑚 𝑆)–1-1-onto→𝐵 ↔ (𝐹 Fn (𝐵 ↑𝑚 𝑆) ∧ ◡𝐹 Fn 𝐵))
16	5, 14, 15	mpbir2an 955	1 ⊢ 𝐹:(𝐵 ↑𝑚 𝑆)–1-1-onto→𝐵

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  Vcvv 2821  {csn 3709   ↦ cmpt 4193   × cxp 4774  ^◡ccnv 4775   Fn wfn 5375  –1-1-onto→wf1o 5379  ‘cfv 5380  (class class class)co 6088   ↑_𝑚 cmap 6926

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-map 6928

This theorem is used by:  mapsnf1o3  6984