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| Mirrors > Home > MPE Home > Th. List > dmmptd | Structured version Visualization version GIF version | ||
| Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| Ref | Expression |
|---|---|
| dmmptd.a | ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶) |
| dmmptd.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) |
| Ref | Expression |
|---|---|
| dmmptd | ⊢ (𝜑 → dom 𝐴 = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmmptd.a | . . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
| 2 | 1 | dmmpt 6260 | . 2 ⊢ dom 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} |
| 3 | dmmptd.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) | |
| 4 | 3 | elexd 3504 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ V) |
| 5 | 4 | ralrimiva 3146 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐶 ∈ V) |
| 6 | rabid2 3470 | . . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑥 ∈ 𝐵 𝐶 ∈ V) | |
| 7 | 5, 6 | sylibr 234 | . 2 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V}) |
| 8 | 2, 7 | eqtr4id 2796 | 1 ⊢ (𝜑 → dom 𝐴 = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ↦ cmpt 5225 dom cdm 5685 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: lo1eq 15604 rlimeq 15605 rlimcld2 15614 rlimcn3 15626 rlimmptrcl 15644 rlimsqzlem 15685 dprdz 20050 alexsublem 24052 cmetcaulem 25322 minveclem3b 25462 mbfneg 25685 mbfsup 25699 mbfinf 25700 mbflimsup 25701 itg2monolem1 25785 itg2mono 25788 itg2i1fseq2 25791 itg2cnlem1 25796 isibl2 25801 iblcnlem 25824 limccnp2 25927 limcco 25928 dvmptres3 25994 itgsubstlem 26089 iblulm 26450 rlimcnp2 27009 dchrisumlema 27532 htthlem 30936 qusrn 33437 algextdeglem4 33761 expgrowth 44354 mptelpm 45181 choicefi 45205 mullimc 45631 limcmptdm 45650 dvsinax 45928 dirkercncflem2 46119 fourierdlem62 46183 psmeasure 46486 ovnovollem2 46672 smfmbfcex 46775 smflimsuplem2 46836 |
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