mapsnf1o2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mapsnf1o2		Structured version Visualization version GIF version

Theorem mapsnf1o2 8458

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 6683	. . 3 ⊢ (𝑥‘𝑋) ∈ V
2		mapsncnv.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋))
3	1, 2	fnmpti 6491	. 2 ⊢ 𝐹 Fn (𝐵 ↑_m 𝑆)
4		mapsncnv.s	. . . . 5 ⊢ 𝑆 = {𝑋}
5		snex 5332	. . . . 5 ⊢ {𝑋} ∈ V
6	4, 5	eqeltri 2909	. . . 4 ⊢ 𝑆 ∈ V
7		snex 5332	. . . 4 ⊢ {𝑦} ∈ V
8	6, 7	xpex 7476	. . 3 ⊢ (𝑆 × {𝑦}) ∈ V
9		mapsncnv.b	. . . 4 ⊢ 𝐵 ∈ V
10		mapsncnv.x	. . . 4 ⊢ 𝑋 ∈ V
11	4, 9, 10, 2	mapsncnv 8457	. . 3 ⊢ ^◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
12	8, 11	fnmpti 6491	. 2 ⊢ ^◡𝐹 Fn 𝐵
13		dff1o4 6623	. 2 ⊢ (𝐹:(𝐵 ↑_m 𝑆)–1-1-onto→𝐵 ↔ (𝐹 Fn (𝐵 ↑_m 𝑆) ∧ ^◡𝐹 Fn 𝐵))
14	3, 12, 13	mpbir2an 709	1 ⊢ 𝐹:(𝐵 ↑_m 𝑆)–1-1-onto→𝐵

Colors of variables: wff setvar class

Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 ↦ cmpt 5146 × cxp 5553 ^◡ccnv 5554 Fn wfn 6350 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 ↑_m cmap 8406

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408

This theorem is referenced by: mapsnf1o3 8459 coe1sfi 20381 coe1mul2lem2 20436 ply1coe 20464 evl1var 20499 pf1mpf 20515 pf1ind 20518 deg1ldg 24686 deg1leb 24689

W3C validator