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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 4639 | . . . . . 6 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎}) | |
2 | 1 | cbvmptv 5262 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑎 ∈ 𝐴 ↦ {𝑎}) |
3 | 2 | eqcomi 2742 | . . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}) = (𝑥 ∈ 𝐴 ↦ {𝑥}) |
4 | id 22 | . . . . . . . 8 ⊢ (𝑢 = 𝑧 → 𝑢 = 𝑧) | |
5 | 4, 1 | eqeqan12d 2747 | . . . . . . 7 ⊢ ((𝑢 = 𝑧 ∧ 𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎})) |
6 | 5 | cbvrexdva 3238 | . . . . . 6 ⊢ (𝑢 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎})) |
7 | 6 | cbvabv 2806 | . . . . 5 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} |
8 | 7 | eqcomi 2742 | . . . 4 ⊢ {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} |
9 | 3, 8 | f1omptsnlem 36217 | . . 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 2761 | . . . 4 ⊢ 𝑅 = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} |
12 | f1oeq3 6824 | . . . 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 230 | . 2 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅 |
15 | f1omptsn.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) | |
16 | 15, 2 | eqtri 2761 | . . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ {𝑎}) |
17 | f1oeq1 6822 | . . 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 230 | 1 ⊢ 𝐹:𝐴–1-1-onto→𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 {cab 2710 ∃wrex 3071 {csn 4629 ↦ cmpt 5232 –1-1-onto→wf1o 6543 |
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 5300 ax-nul 5307 ax-pr 5428 |
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-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 |
This theorem is referenced by: (None) |
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