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Mirrors > Home > MPE Home > Th. List > Mathboxes > f1omptsn | Structured version Visualization version GIF version |
Description: A function mapping to singletons is bijective onto a set of singletons. (Contributed by ML, 16-Jul-2020.) |
Ref | Expression |
---|---|
f1omptsn.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) |
f1omptsn.r | ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} |
Ref | Expression |
---|---|
f1omptsn | ⊢ 𝐹:𝐴–1-1-onto→𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4576 | . . . . . 6 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎}) | |
2 | 1 | cbvmptv 5168 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑎 ∈ 𝐴 ↦ {𝑎}) |
3 | 2 | eqcomi 2830 | . . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}) = (𝑥 ∈ 𝐴 ↦ {𝑥}) |
4 | id 22 | . . . . . . . 8 ⊢ (𝑢 = 𝑧 → 𝑢 = 𝑧) | |
5 | 4, 1 | eqeqan12d 2838 | . . . . . . 7 ⊢ ((𝑢 = 𝑧 ∧ 𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎})) |
6 | 5 | cbvrexdva 3460 | . . . . . 6 ⊢ (𝑢 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎})) |
7 | 6 | cbvabv 2889 | . . . . 5 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} |
8 | 7 | eqcomi 2830 | . . . 4 ⊢ {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} |
9 | 3, 8 | f1omptsnlem 34616 | . . 3 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→{𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} |
10 | f1omptsn.r | . . . . 5 ⊢ 𝑅 = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} | |
11 | 10, 7 | eqtri 2844 | . . . 4 ⊢ 𝑅 = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} |
12 | f1oeq3 6605 | . . . 4 ⊢ (𝑅 = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} → ((𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅 ↔ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→{𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}})) | |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ ((𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅 ↔ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→{𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}}) |
14 | 9, 13 | mpbir 233 | . 2 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅 |
15 | f1omptsn.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) | |
16 | 15, 2 | eqtri 2844 | . . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ {𝑎}) |
17 | f1oeq1 6603 | . . 3 ⊢ (𝐹 = (𝑎 ∈ 𝐴 ↦ {𝑎}) → (𝐹:𝐴–1-1-onto→𝑅 ↔ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅)) | |
18 | 16, 17 | ax-mp 5 | . 2 ⊢ (𝐹:𝐴–1-1-onto→𝑅 ↔ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅) |
19 | 14, 18 | mpbir 233 | 1 ⊢ 𝐹:𝐴–1-1-onto→𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 {cab 2799 ∃wrex 3139 {csn 4566 ↦ cmpt 5145 –1-1-onto→wf1o 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 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-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 |
This theorem is referenced by: (None) |
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