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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 5567 | . . 3 ⊢ Rel (V × V) | |
2 | reldif 5682 | . . 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 33314 | . . 3 ⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))) | |
5 | 4 | releqi 5646 | . 2 ⊢ (Rel Bigcup ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))) |
6 | 3, 5 | mpbir 233 | 1 ⊢ Rel Bigcup |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3494 ∖ cdif 3932 △ csymdif 4217 E cep 5458 × cxp 5547 ran crn 5550 ∘ ccom 5553 Rel wrel 5554 ⊗ ctxp 33286 Bigcup cbigcup 33290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3938 df-in 3942 df-ss 3951 df-opab 5121 df-xp 5555 df-rel 5556 df-bigcup 33314 |
This theorem is referenced by: brbigcup 33354 dfbigcup2 33355 |
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