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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 4883 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} | |
2 | 1 | releqi 5616 | . 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ Rel {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}) |
3 | df-rel 5526 | . 2 ⊢ (Rel {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V)) | |
4 | abss 3988 | . . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V) ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) | |
5 | df-rel 5526 | . . . . . 6 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
6 | dfss2 3901 | . . . . . 6 ⊢ (𝐵 ⊆ (V × V) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) | |
7 | 5, 6 | bitri 278 | . . . . 5 ⊢ (Rel 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) |
8 | 7 | ralbii 3133 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) |
9 | ralcom4 3198 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦(𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ ∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) | |
10 | r19.23v 3238 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) | |
11 | 10 | albii 1821 | . . . 4 ⊢ (∀𝑦∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V)) ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) |
12 | 8, 9, 11 | 3bitri 300 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∀𝑦(∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ (V × V))) |
13 | 4, 12 | bitr4i 281 | . 2 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} ⊆ (V × V) ↔ ∀𝑥 ∈ 𝐴 Rel 𝐵) |
14 | 2, 3, 13 | 3bitri 300 | 1 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 Rel 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1536 ∈ wcel 2111 {cab 2776 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ⊆ wss 3881 ∪ ciun 4881 × cxp 5517 Rel wrel 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-in 3888 df-ss 3898 df-iun 4883 df-rel 5526 |
This theorem is referenced by: reluni 5655 eliunxp 5672 opeliunxp2 5673 dfco2 6065 coiun 6076 fvn0ssdmfun 6819 opeliunxp2f 7859 fsumcom2 15121 fprodcom2 15330 imasaddfnlem 16793 imasvscafn 16802 gsum2d2lem 19086 gsum2d2 19087 gsumcom2 19088 dprd2d2 19159 cnextrel 22668 reldv 24473 dfcnv2 30439 gsumpart 30740 cvmliftlem1 32645 cnviun 40351 coiun1 40353 eliunxp2 44735 |
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