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 3212 | . . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ V) |
5 | 4 | ex 450 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝐶 ∈ V)) |
6 | 1, 5 | ralrimi 2957 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐶 ∈ V) |
7 | rabid2 3118 | . . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑥 ∈ 𝐵 𝐶 ∈ V) | |
8 | 6, 7 | sylibr 224 | . 2 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V}) |
9 | dmmptdf.a | . . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
10 | 9 | dmmpt 5630 | . 2 ⊢ dom 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} |
11 | 8, 10 | syl6reqr 2675 | 1 ⊢ (𝜑 → dom 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 ∀wral 2912 {crab 2916 Vcvv 3200 ↦ cmpt 4729 dom cdm 5114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 |
This theorem is referenced by: smfpimltmpt 40955 smfpimltxrmpt 40967 smfadd 40973 smfpimgtmpt 40989 smfpimgtxrmpt 40992 smfpimioompt 40993 smfrec 40996 smfmul 41002 smfmulc1 41003 smffmpt 41011 smfsupmpt 41021 smfinfmpt 41025 smflimsupmpt 41035 smfliminfmpt 41038 |
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