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Theorem dfbigcup2 35881
Description: Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
dfbigcup2 Bigcup = (𝑥 ∈ V ↦ 𝑥)

Proof of Theorem dfbigcup2
Dummy variables 𝑦 𝑧 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relbigcup 35879 . 2 Rel Bigcup
2 mptrel 5838 . 2 Rel (𝑥 ∈ V ↦ 𝑥)
3 eqcom 2742 . . 3 ( 𝑦 = 𝑧𝑧 = 𝑦)
4 vex 3482 . . . 4 𝑧 ∈ V
54brbigcup 35880 . . 3 (𝑦 Bigcup 𝑧 𝑦 = 𝑧)
6 vex 3482 . . . 4 𝑦 ∈ V
7 eleq1w 2822 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V))
8 unieq 4923 . . . . . . 7 (𝑥 = 𝑦 𝑥 = 𝑦)
98eqeq2d 2746 . . . . . 6 (𝑥 = 𝑦 → (𝑡 = 𝑥𝑡 = 𝑦))
107, 9anbi12d 632 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = 𝑥) ↔ (𝑦 ∈ V ∧ 𝑡 = 𝑦)))
116biantrur 530 . . . . 5 (𝑡 = 𝑦 ↔ (𝑦 ∈ V ∧ 𝑡 = 𝑦))
1210, 11bitr4di 289 . . . 4 (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = 𝑥) ↔ 𝑡 = 𝑦))
13 eqeq1 2739 . . . 4 (𝑡 = 𝑧 → (𝑡 = 𝑦𝑧 = 𝑦))
14 df-mpt 5232 . . . 4 (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ V ∧ 𝑡 = 𝑥)}
156, 4, 12, 13, 14brab 5553 . . 3 (𝑦(𝑥 ∈ V ↦ 𝑥)𝑧𝑧 = 𝑦)
163, 5, 153bitr4i 303 . 2 (𝑦 Bigcup 𝑧𝑦(𝑥 ∈ V ↦ 𝑥)𝑧)
171, 2, 16eqbrriv 5804 1 Bigcup = (𝑥 ∈ V ↦ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  Vcvv 3478   cuni 4912   class class class wbr 5148  cmpt 5231   Bigcup cbigcup 35816
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-symdif 4259  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-bigcup 35840
This theorem is referenced by:  fobigcup  35882
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