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Mirrors > Home > MPE Home > Th. List > reliun | Structured version Visualization version GIF version |
Description: An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008.) |
Ref | Expression |
---|---|
reliun | ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iun 4961 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} | |
2 | 1 | releqi 5738 | . 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ Rel {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}) |
3 | df-rel 5645 | . 2 ⊢ (Rel {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V)) | |
4 | abss 4022 | . . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V) ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) | |
5 | df-rel 5645 | . . . . . 6 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
6 | dfss2 3935 | . . . . . 6 ⊢ (𝐵 ⊆ (V × V) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) | |
7 | 5, 6 | bitri 275 | . . . . 5 ⊢ (Rel 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) |
8 | 7 | ralbii 3097 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) |
9 | ralcom4 3272 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) | |
10 | r19.23v 3180 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) | |
11 | 10 | albii 1822 | . . . 4 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) |
12 | 8, 9, 11 | 3bitri 297 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) |
13 | 4, 12 | bitr4i 278 | . 2 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V) ↔ ∀𝑥 ∈ 𝐴 Rel 𝐵) |
14 | 2, 3, 13 | 3bitri 297 | 1 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 ∈ wcel 2107 {cab 2714 ∀wral 3065 ∃wrex 3074 Vcvv 3448 ⊆ wss 3915 ∪ ciun 4959 × cxp 5636 Rel wrel 5643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-v 3450 df-in 3922 df-ss 3932 df-iun 4961 df-rel 5645 |
This theorem is referenced by: reluni 5779 eliunxp 5798 opeliunxp2 5799 dfco2 6202 coiun 6213 fvn0ssdmfun 7030 opeliunxp2f 8146 fsumcom2 15666 fprodcom2 15874 imasaddfnlem 17417 imasvscafn 17426 gsum2d2lem 19757 gsum2d2 19758 gsumcom2 19759 dprd2d2 19830 cnextrel 23430 reldv 25250 dfcnv2 31634 gsumpart 31939 cvmliftlem1 33919 cnviun 41996 coiun1 41998 eliunxp2 46483 |
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