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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 7031, 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 3444 | . . . 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 2727 | . . . 4 ⊢ {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ 𝑁 ∣ 𝜓} | |
9 | fvmptrab.v | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑁 ∈ V) | |
10 | 8, 9 | rabexd 5329 | . . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑦 ∈ 𝑁 ∣ 𝜓} ∈ V) |
11 | 2, 6, 7, 10 | fvmptd 7006 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
12 | 1 | fvmptndm 7030 | . . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝐹‘𝑋) = ∅) |
13 | df-nel 3042 | . . . 4 ⊢ (𝑋 ∉ 𝑉 ↔ ¬ 𝑋 ∈ 𝑉) | |
14 | fvmptrab.n | . . . . 5 ⊢ (𝑋 ∉ 𝑉 → 𝑁 = ∅) | |
15 | rabeq 3441 | . . . . . 6 ⊢ (𝑁 = ∅ → {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓}) | |
16 | rab0 4378 | . . . . . 6 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅ | |
17 | 15, 16 | eqtr2di 2784 | . . . . 5 ⊢ (𝑁 = ∅ → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
18 | 14, 17 | syl 17 | . . . 4 ⊢ (𝑋 ∉ 𝑉 → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
19 | 13, 18 | sylbir 234 | . . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓}) |
20 | 12, 19 | eqtrd 2767 | . 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 205 = wceq 1534 ∈ wcel 2099 ∉ wnel 3041 {crab 3427 Vcvv 3469 ∅c0 4318 ↦ cmpt 5225 ‘cfv 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 |
This theorem is referenced by: (None) |
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