Theorem dfbigcup2 36389

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.

Step	Hyp	Ref	Expression
1		relbigcup 36387	. 2 ⊢ Rel Bigcup
2		mptrel 5812	. 2 ⊢ Rel (𝑥 ∈ V ↦ ∪ 𝑥)
3		eqcom 2770	. . 3 ⊢ (∪ 𝑦 = 𝑧 ↔ 𝑧 = ∪ 𝑦)
4		vex 3459	. . . 4 ⊢ 𝑧 ∈ V
5	4	brbigcup 36388	. . 3 ⊢ (𝑦 Bigcup 𝑧 ↔ ∪ 𝑦 = 𝑧)
6		vex 3459	. . . 4 ⊢ 𝑦 ∈ V
7		eleq1w 2846	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V))
8		unieq 4883	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ∪ 𝑥 = ∪ 𝑦)
9	8	eqeq2d 2774	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑡 = ∪ 𝑥 ↔ 𝑡 = ∪ 𝑦))
10	7, 9	anbi12d 643	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦)))
11	6	biantrur 539	. . . . 5 ⊢ (𝑡 = ∪ 𝑦 ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦))
12	10, 11	bitr4di 292	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ 𝑡 = ∪ 𝑦))
13		eqeq1 2767	. . . 4 ⊢ (𝑡 = 𝑧 → (𝑡 = ∪ 𝑦 ↔ 𝑧 = ∪ 𝑦))
14		df-mpt 5193	. . . 4 ⊢ (𝑥 ∈ V ↦ ∪ 𝑥) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥)}
15	6, 4, 12, 13, 14	brab 5528	. . 3 ⊢ (𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧 ↔ 𝑧 = ∪ 𝑦)
16	3, 5, 15	3bitr4i 306	. 2 ⊢ (𝑦 Bigcup 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧)
17	1, 2, 16	eqbrriv 5777	1 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∪ cuni 4872   class class class wbr 5109   ↦ cmpt 5192   Bigcup cbigcup 36324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-symdif 4206  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1st 7982  df-2nd 7983  df-txp 36344  df-bigcup 36348

This theorem is referenced by:  fobigcup  36390