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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 33471 | . . 3 ⊢ Rel Bigcup | |
2 | 1 | brrelex1i 5572 | . 2 ⊢ (𝐴 Bigcup 𝐵 → 𝐴 ∈ V) |
3 | brbigcup.1 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | eleq1 2877 | . . . 4 ⊢ (∪ 𝐴 = 𝐵 → (∪ 𝐴 ∈ V ↔ 𝐵 ∈ V)) | |
5 | 3, 4 | mpbiri 261 | . . 3 ⊢ (∪ 𝐴 = 𝐵 → ∪ 𝐴 ∈ V) |
6 | uniexb 7466 | . . 3 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | |
7 | 5, 6 | sylibr 237 | . 2 ⊢ (∪ 𝐴 = 𝐵 → 𝐴 ∈ V) |
8 | breq1 5033 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 Bigcup 𝐵 ↔ 𝐴 Bigcup 𝐵)) | |
9 | unieq 4811 | . . . 4 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
10 | 9 | eqeq1d 2800 | . . 3 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 = 𝐵 ↔ ∪ 𝐴 = 𝐵)) |
11 | vex 3444 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | df-bigcup 33432 | . . . . 5 ⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))) | |
13 | brxp 5565 | . . . . . 6 ⊢ (𝑥(V × V)𝐵 ↔ (𝑥 ∈ V ∧ 𝐵 ∈ V)) | |
14 | 11, 3, 13 | mpbir2an 710 | . . . . 5 ⊢ 𝑥(V × V)𝐵 |
15 | epel 5433 | . . . . . . 7 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
16 | 15 | rexbii 3210 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝑥 𝑦 E 𝑧 ↔ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧) |
17 | vex 3444 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
18 | 17, 11 | coep 33100 | . . . . . 6 ⊢ (𝑦( E ∘ E )𝑥 ↔ ∃𝑧 ∈ 𝑥 𝑦 E 𝑧) |
19 | eluni2 4804 | . . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ↔ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧) | |
20 | 16, 18, 19 | 3bitr4ri 307 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ↔ 𝑦( E ∘ E )𝑥) |
21 | 11, 3, 12, 14, 20 | brtxpsd3 33470 | . . . 4 ⊢ (𝑥 Bigcup 𝐵 ↔ 𝐵 = ∪ 𝑥) |
22 | eqcom 2805 | . . . 4 ⊢ (𝐵 = ∪ 𝑥 ↔ ∪ 𝑥 = 𝐵) | |
23 | 21, 22 | bitri 278 | . . 3 ⊢ (𝑥 Bigcup 𝐵 ↔ ∪ 𝑥 = 𝐵) |
24 | 8, 10, 23 | vtoclbg 3517 | . 2 ⊢ (𝐴 ∈ V → (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵)) |
25 | 2, 7, 24 | pm5.21nii 383 | 1 ⊢ (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ∪ cuni 4800 class class class wbr 5030 E cep 5429 × cxp 5517 ∘ ccom 5523 Bigcup cbigcup 33408 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-symdif 4169 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-eprel 5430 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-1st 7671 df-2nd 7672 df-txp 33428 df-bigcup 33432 |
This theorem is referenced by: dfbigcup2 33473 fvbigcup 33476 ellimits 33484 brapply 33512 dfrdg4 33525 |
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