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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 33358 | . . 3 ⊢ Rel Bigcup | |
2 | 1 | brrelex1i 5607 | . 2 ⊢ (𝐴 Bigcup 𝐵 → 𝐴 ∈ V) |
3 | brbigcup.1 | . . . 4 ⊢ 𝐵 ∈ V | |
4 | eleq1 2900 | . . . 4 ⊢ (∪ 𝐴 = 𝐵 → (∪ 𝐴 ∈ V ↔ 𝐵 ∈ V)) | |
5 | 3, 4 | mpbiri 260 | . . 3 ⊢ (∪ 𝐴 = 𝐵 → ∪ 𝐴 ∈ V) |
6 | uniexb 7485 | . . 3 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V) | |
7 | 5, 6 | sylibr 236 | . 2 ⊢ (∪ 𝐴 = 𝐵 → 𝐴 ∈ V) |
8 | breq1 5068 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 Bigcup 𝐵 ↔ 𝐴 Bigcup 𝐵)) | |
9 | unieq 4848 | . . . 4 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
10 | 9 | eqeq1d 2823 | . . 3 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 = 𝐵 ↔ ∪ 𝐴 = 𝐵)) |
11 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
12 | df-bigcup 33319 | . . . . 5 ⊢ Bigcup = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ E ) ⊗ V))) | |
13 | brxp 5600 | . . . . . 6 ⊢ (𝑥(V × V)𝐵 ↔ (𝑥 ∈ V ∧ 𝐵 ∈ V)) | |
14 | 11, 3, 13 | mpbir2an 709 | . . . . 5 ⊢ 𝑥(V × V)𝐵 |
15 | epel 5468 | . . . . . . 7 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
16 | 15 | rexbii 3247 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝑥 𝑦 E 𝑧 ↔ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧) |
17 | vex 3497 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
18 | 17, 11 | coep 32987 | . . . . . 6 ⊢ (𝑦( E ∘ E )𝑥 ↔ ∃𝑧 ∈ 𝑥 𝑦 E 𝑧) |
19 | eluni2 4841 | . . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑥 ↔ ∃𝑧 ∈ 𝑥 𝑦 ∈ 𝑧) | |
20 | 16, 18, 19 | 3bitr4ri 306 | . . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ↔ 𝑦( E ∘ E )𝑥) |
21 | 11, 3, 12, 14, 20 | brtxpsd3 33357 | . . . 4 ⊢ (𝑥 Bigcup 𝐵 ↔ 𝐵 = ∪ 𝑥) |
22 | eqcom 2828 | . . . 4 ⊢ (𝐵 = ∪ 𝑥 ↔ ∪ 𝑥 = 𝐵) | |
23 | 21, 22 | bitri 277 | . . 3 ⊢ (𝑥 Bigcup 𝐵 ↔ ∪ 𝑥 = 𝐵) |
24 | 8, 10, 23 | vtoclbg 3568 | . 2 ⊢ (𝐴 ∈ V → (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵)) |
25 | 2, 7, 24 | pm5.21nii 382 | 1 ⊢ (𝐴 Bigcup 𝐵 ↔ ∪ 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 Vcvv 3494 ∪ cuni 4837 class class class wbr 5065 E cep 5463 × cxp 5552 ∘ ccom 5558 Bigcup cbigcup 33295 |
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 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-symdif 4218 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-eprel 5464 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fo 6360 df-fv 6362 df-1st 7688 df-2nd 7689 df-txp 33315 df-bigcup 33319 |
This theorem is referenced by: dfbigcup2 33360 fvbigcup 33363 ellimits 33371 brapply 33399 dfrdg4 33412 |
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