Theorem mapsnf1o2 8880

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		fvex 6884	. . 3 ⊢ (𝑥‘𝑋) ∈ V
2		mapsncnv.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋))
3	1, 2	fnmpti 6668	. 2 ⊢ 𝐹 Fn (𝐵 ↑m 𝑆)
4		mapsncnv.s	. . . . 5 ⊢ 𝑆 = {𝑋}
5		snex 5401	. . . . 5 ⊢ {𝑋} ∈ V
6	4, 5	eqeltri 2861	. . . 4 ⊢ 𝑆 ∈ V
7		snex 5401	. . . 4 ⊢ {𝑦} ∈ V
8	6, 7	xpex 7740	. . 3 ⊢ (𝑆 × {𝑦}) ∈ V
9		mapsncnv.b	. . . 4 ⊢ 𝐵 ∈ V
10		mapsncnv.x	. . . 4 ⊢ 𝑋 ∈ V
11	4, 9, 10, 2	mapsncnv 8879	. . 3 ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
12	8, 11	fnmpti 6668	. 2 ⊢ ◡𝐹 Fn 𝐵
13		dff1o4 6819	. 2 ⊢ (𝐹:(𝐵 ↑m 𝑆)–1-1-onto→𝐵 ↔ (𝐹 Fn (𝐵 ↑m 𝑆) ∧ ◡𝐹 Fn 𝐵))
14	3, 12, 13	mpbir2an 723	1 ⊢ 𝐹:(𝐵 ↑m 𝑆)–1-1-onto→𝐵

Syntax hints:   = wceq 1563   ∈ wcel 2145  Vcvv 3457  {csn 4585   ↦ cmpt 5186   × cxp 5650  ^◡ccnv 5651   Fn wfn 6520  –1-1-onto→wf1o 6524  ‘cfv 6525  (class class class)co 7400   ↑_m cmap 8812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814

This theorem is referenced by:  mapsnf1o3  8881  coe1sfi  22333  coe1mul2lem2  22389  ply1coe  22419  evl1var  22457  pf1mpf  22473  pf1ind  22476  deg1ldg  26210  deg1leb  26213