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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 5150 | . . 3 ⊢ Rel (V × V) | |
2 | reldif 5161 | . . 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 31134 | . . 3 ⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))) | |
5 | 4 | releqi 5125 | . 2 ⊢ (Rel Bigcup ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))) |
6 | 3, 5 | mpbir 220 | 1 ⊢ Rel Bigcup |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3173 ∖ cdif 3537 △ csymdif 3805 E cep 4947 × cxp 5036 ran crn 5039 ∘ ccom 5042 Rel wrel 5043 ⊗ ctxp 31106 Bigcup cbigcup 31110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-opab 4644 df-xp 5044 df-rel 5045 df-bigcup 31134 |
This theorem is referenced by: brbigcup 31175 dfbigcup2 31176 |
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