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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fsetsnf | Structured version Visualization version GIF version | ||
| Description: The mapping of an element of a class to a singleton function is a function. (Contributed by AV, 13-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| fsetsnf.a | ⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} | 
| fsetsnf.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {〈𝑆, 𝑥〉}) | 
| Ref | Expression | 
|---|---|
| fsetsnf | ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpr 484 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
| 2 | opeq2 4873 | . . . . . . 7 ⊢ (𝑏 = 𝑥 → 〈𝑆, 𝑏〉 = 〈𝑆, 𝑥〉) | |
| 3 | 2 | sneqd 4637 | . . . . . 6 ⊢ (𝑏 = 𝑥 → {〈𝑆, 𝑏〉} = {〈𝑆, 𝑥〉}) | 
| 4 | 3 | eqeq2d 2747 | . . . . 5 ⊢ (𝑏 = 𝑥 → ({〈𝑆, 𝑥〉} = {〈𝑆, 𝑏〉} ↔ {〈𝑆, 𝑥〉} = {〈𝑆, 𝑥〉})) | 
| 5 | 4 | adantl 481 | . . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ 𝑏 = 𝑥) → ({〈𝑆, 𝑥〉} = {〈𝑆, 𝑏〉} ↔ {〈𝑆, 𝑥〉} = {〈𝑆, 𝑥〉})) | 
| 6 | eqidd 2737 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → {〈𝑆, 𝑥〉} = {〈𝑆, 𝑥〉}) | |
| 7 | 1, 5, 6 | rspcedvd 3623 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → ∃𝑏 ∈ 𝐵 {〈𝑆, 𝑥〉} = {〈𝑆, 𝑏〉}) | 
| 8 | snex 5435 | . . . 4 ⊢ {〈𝑆, 𝑥〉} ∈ V | |
| 9 | eqeq1 2740 | . . . . 5 ⊢ (𝑦 = {〈𝑆, 𝑥〉} → (𝑦 = {〈𝑆, 𝑏〉} ↔ {〈𝑆, 𝑥〉} = {〈𝑆, 𝑏〉})) | |
| 10 | 9 | rexbidv 3178 | . . . 4 ⊢ (𝑦 = {〈𝑆, 𝑥〉} → (∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉} ↔ ∃𝑏 ∈ 𝐵 {〈𝑆, 𝑥〉} = {〈𝑆, 𝑏〉})) | 
| 11 | fsetsnf.a | . . . 4 ⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {〈𝑆, 𝑏〉}} | |
| 12 | 8, 10, 11 | elab2 3681 | . . 3 ⊢ ({〈𝑆, 𝑥〉} ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐵 {〈𝑆, 𝑥〉} = {〈𝑆, 𝑏〉}) | 
| 13 | 7, 12 | sylibr 234 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → {〈𝑆, 𝑥〉} ∈ 𝐴) | 
| 14 | fsetsnf.f | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {〈𝑆, 𝑥〉}) | |
| 15 | 13, 14 | fmptd 7133 | 1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 ∃wrex 3069 {csn 4625 〈cop 4631 ↦ cmpt 5224 ⟶wf 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: fsetsnf1 47069 fsetsnfo 47070 | 
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