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Mirrors > Home > ILE Home > Th. List > elfvmptrab | 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 3063 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑉 → ⦋𝑥 / 𝑚⦌𝑀 = 𝑀) | |
3 | 2 | eqcomd 2176 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 → 𝑀 = ⦋𝑥 / 𝑚⦌𝑀) |
4 | rabeq 2722 | . . . . . 6 ⊢ (𝑀 = ⦋𝑥 / 𝑚⦌𝑀 → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) | |
5 | 3, 4 | syl 14 | . . . . 5 ⊢ (𝑥 ∈ 𝑉 → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) |
6 | 5 | mpteq2ia 4073 | . . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑}) = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) |
7 | 1, 6 | eqtri 2191 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) |
8 | csbconstg 3063 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 = 𝑀) | |
9 | elfvmptrab.v | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑀 ∈ V) | |
10 | 8, 9 | eqeltrd 2247 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V) |
11 | 7, 10 | elfvmptrab1 5588 | . 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)) |
12 | 8 | eleq2d 2240 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀 ↔ 𝑌 ∈ 𝑀)) |
13 | 12 | biimpd 143 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀 → 𝑌 ∈ 𝑀)) |
14 | 13 | imdistani 443 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑀)) |
15 | 11, 14 | syl 14 | 1 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 {crab 2452 Vcvv 2730 ⦋csb 3049 ↦ cmpt 4048 ‘cfv 5196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fv 5204 |
This theorem is referenced by: (None) |
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