Theorem mapsnf1o2 6972

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 5713	. . 3 ⊢ (𝑥‘𝑋) ∈ V
4		mapsncnv.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋))
5	3, 4	fnmpti 5510	. 2 ⊢ 𝐹 Fn (𝐵 ↑𝑚 𝑆)
6		mapsncnv.s	. . . . 5 ⊢ 𝑆 = {𝑋}
7	2	snex 4320	. . . . 5 ⊢ {𝑋} ∈ V
8	6, 7	eqeltri 2311	. . . 4 ⊢ 𝑆 ∈ V
9		vex 2824	. . . . 5 ⊢ 𝑦 ∈ V
10	9	snex 4320	. . . 4 ⊢ {𝑦} ∈ V
11	8, 10	xpex 4889	. . 3 ⊢ (𝑆 × {𝑦}) ∈ V
12		mapsncnv.b	. . . 4 ⊢ 𝐵 ∈ V
13	6, 12, 2, 4	mapsncnv 6971	. . 3 ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
14	11, 13	fnmpti 5510	. 2 ⊢ ◡𝐹 Fn 𝐵
15		dff1o4 5645	. 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

Syntax hints:   = wceq 1402   ∈ wcel 2209  Vcvv 2821  {csn 3708   ↦ cmpt 4190   × cxp 4770  ^◡ccnv 4771   Fn wfn 5370  –1-1-onto→wf1o 5374  ‘cfv 5375  (class class class)co 6079   ↑_𝑚 cmap 6916

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 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 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-map 6918

This theorem is referenced by:  mapsnf1o3  6973