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Mirrors > Home > MPE Home > Th. List > mapsnf1o2 | Structured version Visualization version GIF version |
Description: Explicit bijection between a set and its singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
mapsncnv.s | ⊢ 𝑆 = {𝑋} |
mapsncnv.b | ⊢ 𝐵 ∈ V |
mapsncnv.x | ⊢ 𝑋 ∈ V |
mapsncnv.f | ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) |
Ref | Expression |
---|---|
mapsnf1o2 | ⊢ 𝐹:(𝐵 ↑m 𝑆)–1-1-onto→𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6677 | . . 3 ⊢ (𝑥‘𝑋) ∈ V | |
2 | mapsncnv.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) | |
3 | 1, 2 | fnmpti 6485 | . 2 ⊢ 𝐹 Fn (𝐵 ↑m 𝑆) |
4 | mapsncnv.s | . . . . 5 ⊢ 𝑆 = {𝑋} | |
5 | snex 5323 | . . . . 5 ⊢ {𝑋} ∈ V | |
6 | 4, 5 | eqeltri 2909 | . . . 4 ⊢ 𝑆 ∈ V |
7 | snex 5323 | . . . 4 ⊢ {𝑦} ∈ V | |
8 | 6, 7 | xpex 7464 | . . 3 ⊢ (𝑆 × {𝑦}) ∈ V |
9 | mapsncnv.b | . . . 4 ⊢ 𝐵 ∈ V | |
10 | mapsncnv.x | . . . 4 ⊢ 𝑋 ∈ V | |
11 | 4, 9, 10, 2 | mapsncnv 8446 | . . 3 ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) |
12 | 8, 11 | fnmpti 6485 | . 2 ⊢ ◡𝐹 Fn 𝐵 |
13 | dff1o4 6617 | . 2 ⊢ (𝐹:(𝐵 ↑m 𝑆)–1-1-onto→𝐵 ↔ (𝐹 Fn (𝐵 ↑m 𝑆) ∧ ◡𝐹 Fn 𝐵)) | |
14 | 3, 12, 13 | mpbir2an 707 | 1 ⊢ 𝐹:(𝐵 ↑m 𝑆)–1-1-onto→𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3495 {csn 4559 ↦ cmpt 5138 × cxp 5547 ◡ccnv 5548 Fn wfn 6344 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7145 ↑m cmap 8396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7680 df-2nd 7681 df-map 8398 |
This theorem is referenced by: mapsnf1o3 8448 coe1sfi 20311 coe1mul2lem2 20366 ply1coe 20394 evl1var 20429 pf1mpf 20445 pf1ind 20448 deg1ldg 24615 deg1leb 24618 |
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