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| Mirrors > Home > MPE Home > Th. List > elfvmptrab | Structured version Visualization version GIF version | ||
| Description: Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.) |
| Ref | Expression |
|---|---|
| elfvmptrab.f | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑}) |
| elfvmptrab.v | ⊢ (𝑋 ∈ 𝑉 → 𝑀 ∈ V) |
| Ref | Expression |
|---|---|
| elfvmptrab | ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑀)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfvmptrab.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑}) | |
| 2 | csbconstg 3918 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑉 → ⦋𝑥 / 𝑚⦌𝑀 = 𝑀) | |
| 3 | 2 | eqcomd 2743 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 → 𝑀 = ⦋𝑥 / 𝑚⦌𝑀) |
| 4 | rabeq 3451 | . . . . . 6 ⊢ (𝑀 = ⦋𝑥 / 𝑚⦌𝑀 → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) | |
| 5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝑥 ∈ 𝑉 → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) |
| 6 | 5 | mpteq2ia 5245 | . . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑}) = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) |
| 7 | 1, 6 | eqtri 2765 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) |
| 8 | csbconstg 3918 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 = 𝑀) | |
| 9 | elfvmptrab.v | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑀 ∈ V) | |
| 10 | 8, 9 | eqeltrd 2841 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V) |
| 11 | 7, 10 | elfvmptrab1w 7043 | . 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)) |
| 12 | 8 | eleq2d 2827 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀 ↔ 𝑌 ∈ 𝑀)) |
| 13 | 12 | biimpd 229 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀 → 𝑌 ∈ 𝑀)) |
| 14 | 13 | imdistani 568 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑀)) |
| 15 | 11, 14 | syl 17 | 1 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑀)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ⦋csb 3899 ↦ 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-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-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: (None) |
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