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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 6901 | . . 3 ⊢ (𝑥‘𝑋) ∈ V | |
2 | mapsncnv.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) | |
3 | 1, 2 | fnmpti 6690 | . 2 ⊢ 𝐹 Fn (𝐵 ↑m 𝑆) |
4 | mapsncnv.s | . . . . 5 ⊢ 𝑆 = {𝑋} | |
5 | snex 5430 | . . . . 5 ⊢ {𝑋} ∈ V | |
6 | 4, 5 | eqeltri 2830 | . . . 4 ⊢ 𝑆 ∈ V |
7 | snex 5430 | . . . 4 ⊢ {𝑦} ∈ V | |
8 | 6, 7 | xpex 7735 | . . 3 ⊢ (𝑆 × {𝑦}) ∈ V |
9 | mapsncnv.b | . . . 4 ⊢ 𝐵 ∈ V | |
10 | mapsncnv.x | . . . 4 ⊢ 𝑋 ∈ V | |
11 | 4, 9, 10, 2 | mapsncnv 8883 | . . 3 ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) |
12 | 8, 11 | fnmpti 6690 | . 2 ⊢ ◡𝐹 Fn 𝐵 |
13 | dff1o4 6838 | . 2 ⊢ (𝐹:(𝐵 ↑m 𝑆)–1-1-onto→𝐵 ↔ (𝐹 Fn (𝐵 ↑m 𝑆) ∧ ◡𝐹 Fn 𝐵)) | |
14 | 3, 12, 13 | mpbir2an 710 | 1 ⊢ 𝐹:(𝐵 ↑m 𝑆)–1-1-onto→𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3475 {csn 4627 ↦ cmpt 5230 × cxp 5673 ◡ccnv 5674 Fn wfn 6535 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7404 ↑m cmap 8816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-map 8818 |
This theorem is referenced by: mapsnf1o3 8885 coe1sfi 21719 coe1mul2lem2 21772 ply1coe 21802 evl1var 21837 pf1mpf 21853 pf1ind 21856 deg1ldg 25592 deg1leb 25595 |
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