Theorem dfbigcup2 35887

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 35885	. 2 ⊢ Rel Bigcup
2		mptrel 5788	. 2 ⊢ Rel (𝑥 ∈ V ↦ ∪ 𝑥)
3		eqcom 2736	. . 3 ⊢ (∪ 𝑦 = 𝑧 ↔ 𝑧 = ∪ 𝑦)
4		vex 3451	. . . 4 ⊢ 𝑧 ∈ V
5	4	brbigcup 35886	. . 3 ⊢ (𝑦 Bigcup 𝑧 ↔ ∪ 𝑦 = 𝑧)
6		vex 3451	. . . 4 ⊢ 𝑦 ∈ V
7		eleq1w 2811	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V))
8		unieq 4882	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ∪ 𝑥 = ∪ 𝑦)
9	8	eqeq2d 2740	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑡 = ∪ 𝑥 ↔ 𝑡 = ∪ 𝑦))
10	7, 9	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦)))
11	6	biantrur 530	. . . . 5 ⊢ (𝑡 = ∪ 𝑦 ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦))
12	10, 11	bitr4di 289	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ 𝑡 = ∪ 𝑦))
13		eqeq1 2733	. . . 4 ⊢ (𝑡 = 𝑧 → (𝑡 = ∪ 𝑦 ↔ 𝑧 = ∪ 𝑦))
14		df-mpt 5189	. . . 4 ⊢ (𝑥 ∈ V ↦ ∪ 𝑥) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥)}
15	6, 4, 12, 13, 14	brab 5503	. . 3 ⊢ (𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧 ↔ 𝑧 = ∪ 𝑦)
16	3, 5, 15	3bitr4i 303	. 2 ⊢ (𝑦 Bigcup 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧)
17	1, 2, 16	eqbrriv 5754	1 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ∪ cuni 4871   class class class wbr 5107   ↦ cmpt 5188   Bigcup cbigcup 35822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-symdif 4216  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-eprel 5538  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fo 6517  df-fv 6519  df-1st 7968  df-2nd 7969  df-txp 35842  df-bigcup 35846

This theorem is referenced by:  fobigcup  35888