Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmmptdf | Structured version Visualization version GIF version |
Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
dmmptdf.x | ⊢ Ⅎ𝑥𝜑 |
dmmptdf.a | ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶) |
dmmptdf.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) |
Ref | Expression |
---|---|
dmmptdf | ⊢ (𝜑 → dom 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmmptdf.a | . . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
2 | 1 | dmmpt 6143 | . 2 ⊢ dom 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} |
3 | dmmptdf.x | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | dmmptdf.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) | |
5 | 4 | elexd 3452 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ V) |
6 | 3, 5 | ralrimia 3430 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐶 ∈ V) |
7 | rabid2 3314 | . . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑥 ∈ 𝐵 𝐶 ∈ V) | |
8 | 6, 7 | sylibr 233 | . 2 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V}) |
9 | 2, 8 | eqtr4id 2797 | 1 ⊢ (𝜑 → dom 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 Ⅎwnf 1786 ∈ wcel 2106 ∀wral 3064 {crab 3068 Vcvv 3432 ↦ cmpt 5157 dom cdm 5589 |
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 5223 ax-nul 5230 ax-pr 5352 |
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-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-mpt 5158 df-xp 5595 df-rel 5596 df-cnv 5597 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 |
This theorem is referenced by: smfpimltmpt 44282 smfpimltxrmpt 44294 smfadd 44300 smfpimgtmpt 44316 smfpimgtxrmpt 44319 smfpimioompt 44320 smfrec 44323 smfmul 44329 smfmulc1 44330 smffmpt 44338 smfsupmpt 44348 smfinfmpt 44352 smflimsupmpt 44362 smfliminfmpt 44365 |
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