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Theorem dfbigcup2 33257
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 33255 . 2 Rel Bigcup
2 mptrel 5690 . 2 Rel (𝑥 ∈ V ↦ 𝑥)
3 eqcom 2825 . . 3 ( 𝑦 = 𝑧𝑧 = 𝑦)
4 vex 3495 . . . 4 𝑧 ∈ V
54brbigcup 33256 . . 3 (𝑦 Bigcup 𝑧 𝑦 = 𝑧)
6 vex 3495 . . . 4 𝑦 ∈ V
7 eleq1w 2892 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V))
8 unieq 4838 . . . . . . 7 (𝑥 = 𝑦 𝑥 = 𝑦)
98eqeq2d 2829 . . . . . 6 (𝑥 = 𝑦 → (𝑡 = 𝑥𝑡 = 𝑦))
107, 9anbi12d 630 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = 𝑥) ↔ (𝑦 ∈ V ∧ 𝑡 = 𝑦)))
116biantrur 531 . . . . 5 (𝑡 = 𝑦 ↔ (𝑦 ∈ V ∧ 𝑡 = 𝑦))
1210, 11syl6bbr 290 . . . 4 (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = 𝑥) ↔ 𝑡 = 𝑦))
13 eqeq1 2822 . . . 4 (𝑡 = 𝑧 → (𝑡 = 𝑦𝑧 = 𝑦))
14 df-mpt 5138 . . . 4 (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ V ∧ 𝑡 = 𝑥)}
156, 4, 12, 13, 14brab 5421 . . 3 (𝑦(𝑥 ∈ V ↦ 𝑥)𝑧𝑧 = 𝑦)
163, 5, 153bitr4i 304 . 2 (𝑦 Bigcup 𝑧𝑦(𝑥 ∈ V ↦ 𝑥)𝑧)
171, 2, 16eqbrriv 5657 1 Bigcup = (𝑥 ∈ V ↦ 𝑥)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wcel 2105  Vcvv 3492   cuni 4830   class class class wbr 5057  cmpt 5137   Bigcup cbigcup 33192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-symdif 4216  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-eprel 5458  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7678  df-2nd 7679  df-txp 33212  df-bigcup 33216
This theorem is referenced by:  fobigcup  33258
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