Theorem mapsnf1o2 8933

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 6920	. . 3 ⊢ (𝑥‘𝑋) ∈ V
2		mapsncnv.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋))
3	1, 2	fnmpti 6712	. 2 ⊢ 𝐹 Fn (𝐵 ↑m 𝑆)
4		mapsncnv.s	. . . . 5 ⊢ 𝑆 = {𝑋}
5		snex 5442	. . . . 5 ⊢ {𝑋} ∈ V
6	4, 5	eqeltri 2835	. . . 4 ⊢ 𝑆 ∈ V
7		snex 5442	. . . 4 ⊢ {𝑦} ∈ V
8	6, 7	xpex 7772	. . 3 ⊢ (𝑆 × {𝑦}) ∈ V
9		mapsncnv.b	. . . 4 ⊢ 𝐵 ∈ V
10		mapsncnv.x	. . . 4 ⊢ 𝑋 ∈ V
11	4, 9, 10, 2	mapsncnv 8932	. . 3 ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
12	8, 11	fnmpti 6712	. 2 ⊢ ◡𝐹 Fn 𝐵
13		dff1o4 6857	. 2 ⊢ (𝐹:(𝐵 ↑m 𝑆)–1-1-onto→𝐵 ↔ (𝐹 Fn (𝐵 ↑m 𝑆) ∧ ◡𝐹 Fn 𝐵))
14	3, 12, 13	mpbir2an 711	1 ⊢ 𝐹:(𝐵 ↑m 𝑆)–1-1-onto→𝐵

Syntax hints:   = wceq 1537   ∈ wcel 2106  Vcvv 3478  {csn 4631   ↦ cmpt 5231   × cxp 5687  ^◡ccnv 5688   Fn wfn 6558  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431   ↑_m cmap 8865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867

This theorem is referenced by:  mapsnf1o3  8934  coe1sfi  22231  coe1mul2lem2  22287  ply1coe  22318  evl1var  22356  pf1mpf  22372  pf1ind  22375  deg1ldg  26146  deg1leb  26149