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| Mirrors > Home > MPE Home > Th. List > Mathboxes > brbigcup | Structured version Visualization version GIF version | ||
| Description: Binary relation over Bigcup . (Contributed by Scott Fenton, 11-Apr-2012.) | 
| Ref | Expression | 
|---|---|
| brbigcup.1 | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| brbigcup | ⊢ (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | relbigcup 35898 | . . 3 ⊢ Rel Bigcup | |
| 2 | 1 | brrelex1i 5741 | . 2 ⊢ (𝐴 Bigcup 𝐵 → 𝐴 ∈ V) | 
| 3 | brbigcup.1 | . . . 4 ⊢ 𝐵 ∈ V | |
| 4 | eleq1 2829 | . . . 4 ⊢ (∪ 𝐴 = 𝐵 → (∪ 𝐴 ∈ V ↔ 𝐵 ∈ V)) | |
| 5 | 3, 4 | mpbiri 258 | . . 3 ⊢ (∪ 𝐴 = 𝐵 → ∪ 𝐴 ∈ V) | 
| 6 | uniexb 7784 | . . 3 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | |
| 7 | 5, 6 | sylibr 234 | . 2 ⊢ (∪ 𝐴 = 𝐵 → 𝐴 ∈ V) | 
| 8 | breq1 5146 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 Bigcup 𝐵 ↔ 𝐴 Bigcup 𝐵)) | |
| 9 | unieq 4918 | . . . 4 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
| 10 | 9 | eqeq1d 2739 | . . 3 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 = 𝐵 ↔ ∪ 𝐴 = 𝐵)) | 
| 11 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 12 | df-bigcup 35859 | . . . . 5 ⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))) | |
| 13 | brxp 5734 | . . . . . 6 ⊢ (𝑥(V × V)𝐵 ↔ (𝑥 ∈ V ∧ 𝐵 ∈ V)) | |
| 14 | 11, 3, 13 | mpbir2an 711 | . . . . 5 ⊢ 𝑥(V × V)𝐵 | 
| 15 | epel 5587 | . . . . . . 7 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
| 16 | 15 | rexbii 3094 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝑥 𝑦 E 𝑧 ↔ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧) | 
| 17 | vex 3484 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 18 | 17, 11 | coep 35752 | . . . . . 6 ⊢ (𝑦( E ∘ E )𝑥 ↔ ∃𝑧 ∈ 𝑥 𝑦 E 𝑧) | 
| 19 | eluni2 4911 | . . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ↔ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧) | |
| 20 | 16, 18, 19 | 3bitr4ri 304 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ↔ 𝑦( E ∘ E )𝑥) | 
| 21 | 11, 3, 12, 14, 20 | brtxpsd3 35897 | . . . 4 ⊢ (𝑥 Bigcup 𝐵 ↔ 𝐵 = ∪ 𝑥) | 
| 22 | eqcom 2744 | . . . 4 ⊢ (𝐵 = ∪ 𝑥 ↔ ∪ 𝑥 = 𝐵) | |
| 23 | 21, 22 | bitri 275 | . . 3 ⊢ (𝑥 Bigcup 𝐵 ↔ ∪ 𝑥 = 𝐵) | 
| 24 | 8, 10, 23 | vtoclbg 3557 | . 2 ⊢ (𝐴 ∈ V → (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵)) | 
| 25 | 2, 7, 24 | pm5.21nii 378 | 1 ⊢ (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∪ cuni 4907 class class class wbr 5143 E cep 5583 × cxp 5683 ∘ ccom 5689 Bigcup cbigcup 35835 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-symdif 4253 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-txp 35855 df-bigcup 35859 | 
| This theorem is referenced by: dfbigcup2 35900 fvbigcup 35903 ellimits 35911 brapply 35939 dfrdg4 35952 | 
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