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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 7048, 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 3452 | . . . 4 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
6 | 5 | adantl 481 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
7 | id 22 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝑉) | |
8 | eqid 2735 | . . . 4 ⊢ {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ 𝑁 ∣ 𝜓} | |
9 | fvmptrab.v | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑁 ∈ V) | |
10 | 8, 9 | rabexd 5346 | . . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑦 ∈ 𝑁 ∣ 𝜓} ∈ V) |
11 | 2, 6, 7, 10 | fvmptd 7023 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
12 | 1 | fvmptndm 7047 | . . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝐹‘𝑋) = ∅) |
13 | df-nel 3045 | . . . 4 ⊢ (𝑋 ∉ 𝑉 ↔ ¬ 𝑋 ∈ 𝑉) | |
14 | fvmptrab.n | . . . . 5 ⊢ (𝑋 ∉ 𝑉 → 𝑁 = ∅) | |
15 | rabeq 3448 | . . . . . 6 ⊢ (𝑁 = ∅ → {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓}) | |
16 | rab0 4392 | . . . . . 6 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅ | |
17 | 15, 16 | eqtr2di 2792 | . . . . 5 ⊢ (𝑁 = ∅ → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
18 | 14, 17 | syl 17 | . . . 4 ⊢ (𝑋 ∉ 𝑉 → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
19 | 13, 18 | sylbir 235 | . . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
20 | 12, 19 | eqtrd 2775 | . 2 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
21 | 11, 20 | pm2.61i 182 | 1 ⊢ (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∉ wnel 3044 {crab 3433 Vcvv 3478 ∅c0 4339 ↦ cmpt 5231 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 |
This theorem is referenced by: (None) |
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