Theorem mapsnf1o2 8870

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 6874	. . 3 ⊢ (𝑥‘𝑋) ∈ V
2		mapsncnv.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋))
3	1, 2	fnmpti 6664	. 2 ⊢ 𝐹 Fn (𝐵 ↑m 𝑆)
4		mapsncnv.s	. . . . 5 ⊢ 𝑆 = {𝑋}
5		snex 5394	. . . . 5 ⊢ {𝑋} ∈ V
6	4, 5	eqeltri 2825	. . . 4 ⊢ 𝑆 ∈ V
7		snex 5394	. . . 4 ⊢ {𝑦} ∈ V
8	6, 7	xpex 7732	. . 3 ⊢ (𝑆 × {𝑦}) ∈ V
9		mapsncnv.b	. . . 4 ⊢ 𝐵 ∈ V
10		mapsncnv.x	. . . 4 ⊢ 𝑋 ∈ V
11	4, 9, 10, 2	mapsncnv 8869	. . 3 ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
12	8, 11	fnmpti 6664	. 2 ⊢ ◡𝐹 Fn 𝐵
13		dff1o4 6811	. 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 1540   ∈ wcel 2109  Vcvv 3450  {csn 4592   ↦ cmpt 5191   × cxp 5639  ^◡ccnv 5640   Fn wfn 6509  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804

This theorem is referenced by:  mapsnf1o3  8871  coe1sfi  22105  coe1mul2lem2  22161  ply1coe  22192  evl1var  22230  pf1mpf  22246  pf1ind  22249  deg1ldg  26004  deg1leb  26007