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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvmptrab | Structured version Visualization version GIF version |
Description: Value of a function mapping a set to a class abstraction restricting a class depending on the argument of the function. More general version of fvmptrabfv 6800, but relying on the fact that out-of-domain arguments evaluate to the empty set, which relies on set.mm's particular encoding. (Contributed by AV, 14-Feb-2022.) |
Ref | Expression |
---|---|
fvmptrab.f | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑}) |
fvmptrab.r | ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) |
fvmptrab.s | ⊢ (𝑥 = 𝑋 → 𝑀 = 𝑁) |
fvmptrab.v | ⊢ (𝑋 ∈ 𝑉 → 𝑁 ∈ V) |
fvmptrab.n | ⊢ (𝑋 ∉ 𝑉 → 𝑁 = ∅) |
Ref | Expression |
---|---|
fvmptrab | ⊢ (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptrab.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑}) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑})) |
3 | fvmptrab.s | . . . . 5 ⊢ (𝑥 = 𝑋 → 𝑀 = 𝑁) | |
4 | fvmptrab.r | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | rabeqbidv 3386 | . . . 4 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
6 | 5 | adantl 485 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
7 | id 22 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝑉) | |
8 | eqid 2738 | . . . 4 ⊢ {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ 𝑁 ∣ 𝜓} | |
9 | fvmptrab.v | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑁 ∈ V) | |
10 | 8, 9 | rabexd 5198 | . . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑦 ∈ 𝑁 ∣ 𝜓} ∈ V) |
11 | 2, 6, 7, 10 | fvmptd 6776 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
12 | 1 | fvmptndm 6799 | . . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝐹‘𝑋) = ∅) |
13 | df-nel 3039 | . . . 4 ⊢ (𝑋 ∉ 𝑉 ↔ ¬ 𝑋 ∈ 𝑉) | |
14 | fvmptrab.n | . . . . 5 ⊢ (𝑋 ∉ 𝑉 → 𝑁 = ∅) | |
15 | rabeq 3384 | . . . . . 6 ⊢ (𝑁 = ∅ → {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓}) | |
16 | rab0 4268 | . . . . . 6 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅ | |
17 | 15, 16 | eqtr2di 2790 | . . . . 5 ⊢ (𝑁 = ∅ → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
18 | 14, 17 | syl 17 | . . . 4 ⊢ (𝑋 ∉ 𝑉 → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
19 | 13, 18 | sylbir 238 | . . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
20 | 12, 19 | eqtrd 2773 | . 2 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
21 | 11, 20 | pm2.61i 185 | 1 ⊢ (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1542 ∈ wcel 2113 ∉ wnel 3038 {crab 3057 Vcvv 3397 ∅c0 4209 ↦ cmpt 5107 ‘cfv 6333 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6291 df-fun 6335 df-fv 6341 |
This theorem is referenced by: (None) |
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