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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.x | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | dmmptdf.c | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) | |
3 | elex 3399 | . . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ V) |
5 | 4 | ex 402 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝐶 ∈ V)) |
6 | 1, 5 | ralrimi 3137 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐶 ∈ V) |
7 | rabid2 3299 | . . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑥 ∈ 𝐵 𝐶 ∈ V) | |
8 | 6, 7 | sylibr 226 | . 2 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V}) |
9 | dmmptdf.a | . . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
10 | 9 | dmmpt 5848 | . 2 ⊢ dom 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} |
11 | 8, 10 | syl6reqr 2851 | 1 ⊢ (𝜑 → dom 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 Ⅎwnf 1879 ∈ wcel 2157 ∀wral 3088 {crab 3092 Vcvv 3384 ↦ cmpt 4921 dom cdm 5311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pr 5096 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ral 3093 df-rab 3097 df-v 3386 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-nul 4115 df-if 4277 df-sn 4368 df-pr 4370 df-op 4374 df-br 4843 df-opab 4905 df-mpt 4922 df-xp 5317 df-rel 5318 df-cnv 5319 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 |
This theorem is referenced by: smfpimltmpt 41690 smfpimltxrmpt 41702 smfadd 41708 smfpimgtmpt 41724 smfpimgtxrmpt 41727 smfpimioompt 41728 smfrec 41731 smfmul 41737 smfmulc1 41738 smffmpt 41746 smfsupmpt 41756 smfinfmpt 41760 smflimsupmpt 41770 smfliminfmpt 41773 |
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