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Mirrors > Home > MPE Home > Th. List > Mathboxes > relbigcup | Structured version Visualization version GIF version |
Description: The Bigcup relationship is a relationship. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
relbigcup | ⊢ Rel Bigcup |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 5694 | . . 3 ⊢ Rel (V × V) | |
2 | reldif 5815 | . . 3 ⊢ (Rel (V × V) → Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))) |
4 | df-bigcup 34899 | . . 3 ⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))) | |
5 | 4 | releqi 5777 | . 2 ⊢ (Rel Bigcup ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))) |
6 | 3, 5 | mpbir 230 | 1 ⊢ Rel Bigcup |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3474 ∖ cdif 3945 △ csymdif 4241 E cep 5579 × cxp 5674 ran crn 5677 ∘ ccom 5680 Rel wrel 5681 ⊗ ctxp 34871 Bigcup cbigcup 34875 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-dif 3951 df-in 3955 df-ss 3965 df-opab 5211 df-xp 5682 df-rel 5683 df-bigcup 34899 |
This theorem is referenced by: brbigcup 34939 dfbigcup2 34940 |
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