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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 4187 | . . . . . 6 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎}) | |
2 | 1 | cbvmptv 4750 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑎 ∈ 𝐴 ↦ {𝑎}) |
3 | 2 | eqcomi 2631 | . . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}) = (𝑥 ∈ 𝐴 ↦ {𝑥}) |
4 | id 22 | . . . . . . . 8 ⊢ (𝑢 = 𝑧 → 𝑢 = 𝑧) | |
5 | 4, 1 | eqeqan12d 2638 | . . . . . . 7 ⊢ ((𝑢 = 𝑧 ∧ 𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎})) |
6 | 5 | cbvrexdva 3178 | . . . . . 6 ⊢ (𝑢 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎})) |
7 | 6 | cbvabv 2747 | . . . . 5 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} |
8 | 7 | eqcomi 2631 | . . . 4 ⊢ {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} |
9 | 3, 8 | f1omptsnlem 33183 | . . 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 2644 | . . . 4 ⊢ 𝑅 = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} |
12 | f1oeq3 6129 | . . . 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 221 | . 2 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅 |
15 | f1omptsn.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) | |
16 | 15, 2 | eqtri 2644 | . . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ {𝑎}) |
17 | f1oeq1 6127 | . . 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 221 | 1 ⊢ 𝐹:𝐴–1-1-onto→𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1483 {cab 2608 ∃wrex 2913 {csn 4177 ↦ cmpt 4729 –1-1-onto→wf1o 5887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 |
This theorem is referenced by: (None) |
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