Theorem dfbigcup2 36110

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 36108	. 2 ⊢ Rel Bigcup
2		mptrel 5782	. 2 ⊢ Rel (𝑥 ∈ V ↦ ∪ 𝑥)
3		eqcom 2744	. . 3 ⊢ (∪ 𝑦 = 𝑧 ↔ 𝑧 = ∪ 𝑦)
4		vex 3446	. . . 4 ⊢ 𝑧 ∈ V
5	4	brbigcup 36109	. . 3 ⊢ (𝑦 Bigcup 𝑧 ↔ ∪ 𝑦 = 𝑧)
6		vex 3446	. . . 4 ⊢ 𝑦 ∈ V
7		eleq1w 2820	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V))
8		unieq 4876	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ∪ 𝑥 = ∪ 𝑦)
9	8	eqeq2d 2748	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑡 = ∪ 𝑥 ↔ 𝑡 = ∪ 𝑦))
10	7, 9	anbi12d 633	. . . . 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 2741	. . . 4 ⊢ (𝑡 = 𝑧 → (𝑡 = ∪ 𝑦 ↔ 𝑧 = ∪ 𝑦))
14		df-mpt 5182	. . . 4 ⊢ (𝑥 ∈ V ↦ ∪ 𝑥) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥)}
15	6, 4, 12, 13, 14	brab 5499	. . 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 5748	1 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∪ cuni 4865   class class class wbr 5100   ↦ cmpt 5181   Bigcup cbigcup 36045

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-symdif 4207  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-eprel 5532  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-1st 7943  df-2nd 7944  df-txp 36065  df-bigcup 36069

This theorem is referenced by:  fobigcup  36111