Mathbox for Alexander van der Vekens |
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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 488 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
2 | opeq2 4767 | . . . . . . 7 ⊢ (𝑏 = 𝑥 → 〈𝑆, 𝑏〉 = 〈𝑆, 𝑥〉) | |
3 | 2 | sneqd 4538 | . . . . . 6 ⊢ (𝑏 = 𝑥 → {〈𝑆, 𝑏〉} = {〈𝑆, 𝑥〉}) |
4 | 3 | eqeq2d 2770 | . . . . 5 ⊢ (𝑏 = 𝑥 → ({〈𝑆, 𝑥〉} = {〈𝑆, 𝑏〉} ↔ {〈𝑆, 𝑥〉} = {〈𝑆, 𝑥〉})) |
5 | 4 | adantl 485 | . . . 4 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) ∧ 𝑏 = 𝑥) → ({〈𝑆, 𝑥〉} = {〈𝑆, 𝑏〉} ↔ {〈𝑆, 𝑥〉} = {〈𝑆, 𝑥〉})) |
6 | eqidd 2760 | . . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → {〈𝑆, 𝑥〉} = {〈𝑆, 𝑥〉}) | |
7 | 1, 5, 6 | rspcedvd 3547 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → ∃𝑏 ∈ 𝐵 {〈𝑆, 𝑥〉} = {〈𝑆, 𝑏〉}) |
8 | snex 5305 | . . . 4 ⊢ {〈𝑆, 𝑥〉} ∈ V | |
9 | eqeq1 2763 | . . . . 5 ⊢ (𝑦 = {〈𝑆, 𝑥〉} → (𝑦 = {〈𝑆, 𝑏〉} ↔ {〈𝑆, 𝑥〉} = {〈𝑆, 𝑏〉})) | |
10 | 9 | rexbidv 3222 | . . . 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 3594 | . . 3 ⊢ ({〈𝑆, 𝑥〉} ∈ 𝐴 ↔ ∃𝑏 ∈ 𝐵 {〈𝑆, 𝑥〉} = {〈𝑆, 𝑏〉}) |
13 | 7, 12 | sylibr 237 | . 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → {〈𝑆, 𝑥〉} ∈ 𝐴) |
14 | fsetsnf.f | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {〈𝑆, 𝑥〉}) | |
15 | 13, 14 | fmptd 6876 | 1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1539 ∈ wcel 2112 {cab 2736 ∃wrex 3072 {csn 4526 〈cop 4532 ↦ cmpt 5117 ⟶wf 6337 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5174 ax-nul 5181 ax-pr 5303 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ral 3076 df-rex 3077 df-rab 3080 df-v 3412 df-sbc 3700 df-dif 3864 df-un 3866 df-in 3868 df-ss 3878 df-nul 4229 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4803 df-br 5038 df-opab 5100 df-mpt 5118 df-id 5435 df-xp 5535 df-rel 5536 df-cnv 5537 df-co 5538 df-dm 5539 df-rn 5540 df-res 5541 df-ima 5542 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-fv 6349 |
This theorem is referenced by: fsetsnf1 44055 fsetsnfo 44056 |
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