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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 6142 | . 2 ⊢ dom 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} |
3 | dmmptd.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) | |
4 | 3 | elexd 3451 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ V) |
5 | 4 | ralrimiva 3110 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐶 ∈ V) |
6 | rabid2 3313 | . . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑥 ∈ 𝐵 𝐶 ∈ V) | |
7 | 5, 6 | sylibr 233 | . 2 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V}) |
8 | 2, 7 | eqtr4id 2799 | 1 ⊢ (𝜑 → dom 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 {crab 3070 Vcvv 3431 ↦ cmpt 5162 dom cdm 5590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-br 5080 df-opab 5142 df-mpt 5163 df-xp 5596 df-rel 5597 df-cnv 5598 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 |
This theorem is referenced by: lo1eq 15275 rlimeq 15276 rlimcld2 15285 rlimcn3 15297 rlimmptrcl 15315 rlimsqzlem 15358 dprdz 19631 alexsublem 23193 cmetcaulem 24450 minveclem3b 24590 mbfneg 24812 mbfsup 24826 mbfinf 24827 mbflimsup 24828 itg2monolem1 24913 itg2mono 24916 itg2i1fseq2 24919 itg2cnlem1 24924 isibl2 24929 iblcnlem 24951 limccnp2 25054 limcco 25055 dvmptres3 25118 itgsubstlem 25210 iblulm 25564 rlimcnp2 26114 dchrisumlema 26634 htthlem 29275 expgrowth 41923 mptelpm 42682 choicefi 42710 mullimc 43128 limcmptdm 43147 dvsinax 43425 dirkercncflem2 43616 fourierdlem62 43680 psmeasure 43980 ovnovollem2 44166 smfmbfcex 44263 smflimsuplem2 44322 |
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