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Mirrors > Home > MPE Home > Th. List > dfiin2 | Structured version Visualization version GIF version |
Description: Alternate definition of indexed intersection when 𝐵 is a set. Definition 15(b) of [Suppes] p. 44. (Contributed by NM, 28-Jun-1998.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
dfiun2.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
dfiin2 | ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiin2g 4959 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) | |
2 | dfiun2.1 | . . 3 ⊢ 𝐵 ∈ V | |
3 | 2 | a1i 11 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ V) |
4 | 1, 3 | mprg 3154 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 {cab 2801 ∃wrex 3141 Vcvv 3496 ∩ cint 4878 ∩ ciin 4922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-v 3498 df-int 4879 df-iin 4924 |
This theorem is referenced by: fniinfv 6744 scott0 9317 cfval2 9684 cflim3 9686 cflim2 9687 cfss 9689 hauscmplem 22016 ptbasfi 22191 dihglblem5 38436 dihglb2 38480 intima0 39999 |
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