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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 5227 | . . 3 ⊢ Rel (V × V) | |
2 | reldif 5238 | . . 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 31965 | . . 3 ⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))) | |
5 | 4 | releqi 5202 | . 2 ⊢ (Rel Bigcup ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V)))) |
6 | 3, 5 | mpbir 221 | 1 ⊢ Rel Bigcup |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3200 ∖ cdif 3571 △ csymdif 3843 E cep 5028 × cxp 5112 ran crn 5115 ∘ ccom 5118 Rel wrel 5119 ⊗ ctxp 31937 Bigcup cbigcup 31941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-opab 4713 df-xp 5120 df-rel 5121 df-bigcup 31965 |
This theorem is referenced by: brbigcup 32005 dfbigcup2 32006 |
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