Theorem dfbigcup2 36095

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 36093	. 2 ⊢ Rel Bigcup
2		mptrel 5774	. 2 ⊢ Rel (𝑥 ∈ V ↦ ∪ 𝑥)
3		eqcom 2744	. . 3 ⊢ (∪ 𝑦 = 𝑧 ↔ 𝑧 = ∪ 𝑦)
4		vex 3434	. . . 4 ⊢ 𝑧 ∈ V
5	4	brbigcup 36094	. . 3 ⊢ (𝑦 Bigcup 𝑧 ↔ ∪ 𝑦 = 𝑧)
6		vex 3434	. . . 4 ⊢ 𝑦 ∈ V
7		eleq1w 2820	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V))
8		unieq 4862	. . . . . . 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 5168	. . . 4 ⊢ (𝑥 ∈ V ↦ ∪ 𝑥) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥)}
15	6, 4, 12, 13, 14	brab 5491	. . 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 5740	1 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ∪ cuni 4851   class class class wbr 5086   ↦ cmpt 5167   Bigcup cbigcup 36030

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 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-symdif 4194  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-eprel 5524  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7935  df-2nd 7936  df-txp 36050  df-bigcup 36054

This theorem is referenced by:  fobigcup  36096