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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 3455 | . . . 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 2737 | . . . 4 ⊢ {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ 𝑁 ∣ 𝜓} | |
| 9 | fvmptrab.v | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑁 ∈ V) | |
| 10 | 8, 9 | rabexd 5340 | . . 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 3047 | . . . 4 ⊢ (𝑋 ∉ 𝑉 ↔ ¬ 𝑋 ∈ 𝑉) | |
| 14 | fvmptrab.n | . . . . 5 ⊢ (𝑋 ∉ 𝑉 → 𝑁 = ∅) | |
| 15 | rabeq 3451 | . . . . . 6 ⊢ (𝑁 = ∅ → {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓}) | |
| 16 | rab0 4386 | . . . . . 6 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅ | |
| 17 | 15, 16 | eqtr2di 2794 | . . . . 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 2777 | . 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 1540 ∈ wcel 2108 ∉ wnel 3046 {crab 3436 Vcvv 3480 ∅c0 4333 ↦ cmpt 5225 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: (None) |
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