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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvmpt2bd | Structured version Visualization version GIF version |
Description: Value of a function given by the maps-to notation. Deduction version. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fvmpt2bd.1 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
Ref | Expression |
---|---|
fvmpt2bd | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝑥) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmpt2bd.1 | . . . 4 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
2 | 1 | fveq1d 6768 | . . 3 ⊢ (𝜑 → (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) |
3 | 2 | 3ad2ant1 1132 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) |
4 | eqid 2738 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
5 | 4 | fvmpt2 6878 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
6 | 5 | 3adant1 1129 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
7 | 3, 6 | eqtrd 2778 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝑥) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ↦ cmpt 5156 ‘cfv 6426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fv 6434 |
This theorem is referenced by: (None) |
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