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| 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 5031 | . 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 3066 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 {cab 2713 ∃wrex 3069 Vcvv 3479 ∩ cint 4945 ∩ ciin 4991 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-v 3481 df-int 4946 df-iin 4993 | 
| This theorem is referenced by: fniinfv 6986 scott0 9927 cfval2 10301 cflim3 10303 cflim2 10304 cfss 10306 hauscmplem 23415 ptbasfi 23590 dihglblem5 41301 dihglb2 41345 intima0 43666 | 
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