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Theorem dfbigcup2 34451
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 34449 . 2 Rel Bigcup
2 mptrel 5779 . 2 Rel (𝑥 ∈ V ↦ 𝑥)
3 eqcom 2743 . . 3 ( 𝑦 = 𝑧𝑧 = 𝑦)
4 vex 3447 . . . 4 𝑧 ∈ V
54brbigcup 34450 . . 3 (𝑦 Bigcup 𝑧 𝑦 = 𝑧)
6 vex 3447 . . . 4 𝑦 ∈ V
7 eleq1w 2820 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V))
8 unieq 4874 . . . . . . 7 (𝑥 = 𝑦 𝑥 = 𝑦)
98eqeq2d 2747 . . . . . 6 (𝑥 = 𝑦 → (𝑡 = 𝑥𝑡 = 𝑦))
107, 9anbi12d 631 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = 𝑥) ↔ (𝑦 ∈ V ∧ 𝑡 = 𝑦)))
116biantrur 531 . . . . 5 (𝑡 = 𝑦 ↔ (𝑦 ∈ V ∧ 𝑡 = 𝑦))
1210, 11bitr4di 288 . . . 4 (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = 𝑥) ↔ 𝑡 = 𝑦))
13 eqeq1 2740 . . . 4 (𝑡 = 𝑧 → (𝑡 = 𝑦𝑧 = 𝑦))
14 df-mpt 5187 . . . 4 (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ V ∧ 𝑡 = 𝑥)}
156, 4, 12, 13, 14brab 5498 . . 3 (𝑦(𝑥 ∈ V ↦ 𝑥)𝑧𝑧 = 𝑦)
163, 5, 153bitr4i 302 . 2 (𝑦 Bigcup 𝑧𝑦(𝑥 ∈ V ↦ 𝑥)𝑧)
171, 2, 16eqbrriv 5745 1 Bigcup = (𝑥 ∈ V ↦ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  Vcvv 3443   cuni 4863   class class class wbr 5103  cmpt 5186   Bigcup cbigcup 34386
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-symdif 4200  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-eprel 5535  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7917  df-2nd 7918  df-txp 34406  df-bigcup 34410
This theorem is referenced by:  fobigcup  34452
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