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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 4565 | . . . . . 6 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎}) | |
2 | 1 | cbvmptv 5172 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑎 ∈ 𝐴 ↦ {𝑎}) |
3 | 2 | eqcomi 2747 | . . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}) = (𝑥 ∈ 𝐴 ↦ {𝑥}) |
4 | id 22 | . . . . . . . 8 ⊢ (𝑢 = 𝑧 → 𝑢 = 𝑧) | |
5 | 4, 1 | eqeqan12d 2752 | . . . . . . 7 ⊢ ((𝑢 = 𝑧 ∧ 𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎})) |
6 | 5 | cbvrexdva 3382 | . . . . . 6 ⊢ (𝑢 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎})) |
7 | 6 | cbvabv 2812 | . . . . 5 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} |
8 | 7 | eqcomi 2747 | . . . 4 ⊢ {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} |
9 | 3, 8 | f1omptsnlem 35270 | . . 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 2766 | . . . 4 ⊢ 𝑅 = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} |
12 | f1oeq3 6669 | . . . 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 234 | . 2 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}):𝐴–1-1-onto→𝑅 |
15 | f1omptsn.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ {𝑥}) | |
16 | 15, 2 | eqtri 2766 | . . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ {𝑎}) |
17 | f1oeq1 6667 | . . 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 234 | 1 ⊢ 𝐹:𝐴–1-1-onto→𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 {cab 2715 ∃wrex 3063 {csn 4555 ↦ cmpt 5149 –1-1-onto→wf1o 6396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-mpt 5150 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 |
This theorem is referenced by: (None) |
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