![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dfiin3 | Structured version Visualization version GIF version |
Description: Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
dfiun3.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
dfiin3 | ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiin3g 5972 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
2 | dfiun3.1 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 2 | a1i 11 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ V) |
4 | 1, 3 | mprg 3057 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∩ cint 4954 ∩ ciin 5002 ↦ cmpt 5236 ran crn 5683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-int 4955 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-cnv 5690 df-dm 5692 df-rn 5693 |
This theorem is referenced by: subdrgint 20782 fclscmpi 24024 |
Copyright terms: Public domain | W3C validator |