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Mirrors > Home > ILE Home > Th. List > mapsnf1o2 | 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 | vex 2724 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | mapsncnv.x | . . . 4 ⊢ 𝑋 ∈ V | |
3 | 1, 2 | fvex 5500 | . . 3 ⊢ (𝑥‘𝑋) ∈ V |
4 | mapsncnv.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)) | |
5 | 3, 4 | fnmpti 5310 | . 2 ⊢ 𝐹 Fn (𝐵 ↑𝑚 𝑆) |
6 | mapsncnv.s | . . . . 5 ⊢ 𝑆 = {𝑋} | |
7 | 2 | snex 4158 | . . . . 5 ⊢ {𝑋} ∈ V |
8 | 6, 7 | eqeltri 2237 | . . . 4 ⊢ 𝑆 ∈ V |
9 | vex 2724 | . . . . 5 ⊢ 𝑦 ∈ V | |
10 | 9 | snex 4158 | . . . 4 ⊢ {𝑦} ∈ V |
11 | 8, 10 | xpex 4713 | . . 3 ⊢ (𝑆 × {𝑦}) ∈ V |
12 | mapsncnv.b | . . . 4 ⊢ 𝐵 ∈ V | |
13 | 6, 12, 2, 4 | mapsncnv 6652 | . . 3 ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) |
14 | 11, 13 | fnmpti 5310 | . 2 ⊢ ◡𝐹 Fn 𝐵 |
15 | dff1o4 5434 | . 2 ⊢ (𝐹:(𝐵 ↑𝑚 𝑆)–1-1-onto→𝐵 ↔ (𝐹 Fn (𝐵 ↑𝑚 𝑆) ∧ ◡𝐹 Fn 𝐵)) | |
16 | 5, 14, 15 | mpbir2an 931 | 1 ⊢ 𝐹:(𝐵 ↑𝑚 𝑆)–1-1-onto→𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∈ wcel 2135 Vcvv 2721 {csn 3570 ↦ cmpt 4037 × cxp 4596 ◡ccnv 4597 Fn wfn 5177 –1-1-onto→wf1o 5181 ‘cfv 5182 (class class class)co 5836 ↑𝑚 cmap 6605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-map 6607 |
This theorem is referenced by: mapsnf1o3 6654 |
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