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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 3905 | . . . . . . 7 ⊢ (𝑥 ∈ 𝑉 → ⦋𝑥 / 𝑚⦌𝑀 = 𝑀) | |
3 | 2 | eqcomd 2830 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 → 𝑀 = ⦋𝑥 / 𝑚⦌𝑀) |
4 | rabeq 3486 | . . . . . 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 5160 | . . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑}) = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) |
7 | 1, 6 | eqtri 2847 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) |
8 | csbconstg 3905 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 = 𝑀) | |
9 | elfvmptrab.v | . . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑀 ∈ V) | |
10 | 8, 9 | eqeltrd 2916 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V) |
11 | 7, 10 | elfvmptrab1w 6797 | . 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)) |
12 | 8 | eleq2d 2901 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀 ↔ 𝑌 ∈ 𝑀)) |
13 | 12 | biimpd 231 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀 → 𝑌 ∈ 𝑀)) |
14 | 13 | imdistani 571 | . 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 398 = wceq 1536 ∈ wcel 2113 {crab 3145 Vcvv 3497 ⦋csb 3886 ↦ cmpt 5149 ‘cfv 6358 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fv 6366 |
This theorem is referenced by: (None) |
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