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Theorem dfbigcup2 31645
 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 31643 . 2 Rel Bigcup
2 mptrel 5208 . 2 Rel (𝑥 ∈ V ↦ 𝑥)
3 eqcom 2628 . . 3 ( 𝑦 = 𝑧𝑧 = 𝑦)
4 vex 3189 . . . 4 𝑧 ∈ V
54brbigcup 31644 . . 3 (𝑦 Bigcup 𝑧 𝑦 = 𝑧)
6 vex 3189 . . . 4 𝑦 ∈ V
7 eleq1 2686 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V))
8 unieq 4410 . . . . . . 7 (𝑥 = 𝑦 𝑥 = 𝑦)
98eqeq2d 2631 . . . . . 6 (𝑥 = 𝑦 → (𝑡 = 𝑥𝑡 = 𝑦))
107, 9anbi12d 746 . . . . 5 (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = 𝑥) ↔ (𝑦 ∈ V ∧ 𝑡 = 𝑦)))
116biantrur 527 . . . . 5 (𝑡 = 𝑦 ↔ (𝑦 ∈ V ∧ 𝑡 = 𝑦))
1210, 11syl6bbr 278 . . . 4 (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = 𝑥) ↔ 𝑡 = 𝑦))
13 eqeq1 2625 . . . 4 (𝑡 = 𝑧 → (𝑡 = 𝑦𝑧 = 𝑦))
14 df-mpt 4675 . . . 4 (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ V ∧ 𝑡 = 𝑥)}
156, 4, 12, 13, 14brab 4958 . . 3 (𝑦(𝑥 ∈ V ↦ 𝑥)𝑧𝑧 = 𝑦)
163, 5, 153bitr4i 292 . 2 (𝑦 Bigcup 𝑧𝑦(𝑥 ∈ V ↦ 𝑥)𝑧)
171, 2, 16eqbrriv 5176 1 Bigcup = (𝑥 ∈ V ↦ 𝑥)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186  ∪ cuni 4402   class class class wbr 4613   ↦ cmpt 4673   Bigcup cbigcup 31579 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-symdif 3822  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-eprel 4985  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-1st 7113  df-2nd 7114  df-txp 31599  df-bigcup 31603 This theorem is referenced by:  fobigcup  31646
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