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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 4580 | . . . . . 6 ⊢ (𝑥 = 𝑎 → {𝑥} = {𝑎}) | |
2 | 1 | cbvmptv 5172 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ {𝑥}) = (𝑎 ∈ 𝐴 ↦ {𝑎}) |
3 | 2 | eqcomi 2833 | . . . 4 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎}) = (𝑥 ∈ 𝐴 ↦ {𝑥}) |
4 | id 22 | . . . . . . . 8 ⊢ (𝑢 = 𝑧 → 𝑢 = 𝑧) | |
5 | 4, 1 | eqeqan12d 2841 | . . . . . . 7 ⊢ ((𝑢 = 𝑧 ∧ 𝑥 = 𝑎) → (𝑢 = {𝑥} ↔ 𝑧 = {𝑎})) |
6 | 5 | cbvrexdva 3463 | . . . . . 6 ⊢ (𝑢 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑢 = {𝑥} ↔ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎})) |
7 | 6 | cbvabv 2892 | . . . . 5 ⊢ {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} |
8 | 7 | eqcomi 2833 | . . . 4 ⊢ {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} = {𝑢 ∣ ∃𝑥 ∈ 𝐴 𝑢 = {𝑥}} |
9 | 3, 8 | f1omptsnlem 34621 | . . 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 2847 | . . . 4 ⊢ 𝑅 = {𝑧 ∣ ∃𝑎 ∈ 𝐴 𝑧 = {𝑎}} |
12 | f1oeq3 6609 | . . . 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 2847 | . . 3 ⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ {𝑎}) |
17 | f1oeq1 6607 | . . 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 1536 {cab 2802 ∃wrex 3142 {csn 4570 ↦ cmpt 5149 –1-1-onto→wf1o 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 |
This theorem is referenced by: (None) |
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