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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmmptdff | Structured version Visualization version GIF version |
Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 21-Dec-2024.) |
Ref | Expression |
---|---|
dmmptdff.x | ⊢ Ⅎ𝑥𝜑 |
dmmptdff.1 | ⊢ Ⅎ𝑥𝐵 |
dmmptdff.a | ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶) |
dmmptdff.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) |
Ref | Expression |
---|---|
dmmptdff | ⊢ (𝜑 → dom 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmmptdff.a | . . 3 ⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶) | |
2 | 1 | dmmpt 6228 | . 2 ⊢ dom 𝐴 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} |
3 | dmmptdff.x | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | dmmptdff.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉) | |
5 | 4 | elexd 3493 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ V) |
6 | 3, 5 | ralrimia 3254 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 𝐶 ∈ V) |
7 | dmmptdff.1 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
8 | 7 | rabid2f 3463 | . . 3 ⊢ (𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V} ↔ ∀𝑥 ∈ 𝐵 𝐶 ∈ V) |
9 | 6, 8 | sylibr 233 | . 2 ⊢ (𝜑 → 𝐵 = {𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V}) |
10 | 2, 9 | eqtr4id 2790 | 1 ⊢ (𝜑 → dom 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2882 ∀wral 3060 {crab 3431 Vcvv 3473 ↦ cmpt 5224 dom cdm 5669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-mpt 5225 df-xp 5675 df-rel 5676 df-cnv 5677 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 |
This theorem is referenced by: dmmptdf 43694 dmmpt1 43746 adddmmbl 45322 muldmmbl 45324 fsupdm2 45332 finfdm2 45336 |
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