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| Mirrors > Home > MPE Home > Th. List > Mathboxes > iinexd | Structured version Visualization version GIF version | ||
| Description: The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by Glauco Siliprandi, 23-Oct-2021.) | 
| Ref | Expression | 
|---|---|
| iinexd.1 | ⊢ (𝜑 → 𝐴 ≠ ∅) | 
| iinexd.2 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) | 
| Ref | Expression | 
|---|---|
| iinexd | ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | iinexd.1 | . 2 ⊢ (𝜑 → 𝐴 ≠ ∅) | |
| 2 | iinexd.2 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) | |
| 3 | iinexg 5348 | . 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ∅c0 4333 ∩ ciin 4992 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 df-ss 3968 df-nul 4334 df-int 4947 df-iin 4994 | 
| This theorem is referenced by: smfsuplem1 46826 smfinflem 46832 smflimsuplem1 46835 smflimsuplem2 46836 smflimsuplem3 46837 smflimsuplem4 46838 smflimsuplem5 46839 smflimsuplem7 46841 | 
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