Theorem dfbigcup2 36641

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 36639	. 2 ⊢ Rel Bigcup
2		mptrel 5803	. 2 ⊢ Rel (𝑥 ∈ V ↦ ∪ 𝑥)
3		eqcom 2768	. . 3 ⊢ (∪ 𝑦 = 𝑧 ↔ 𝑧 = ∪ 𝑦)
4		vex 3455	. . . 4 ⊢ 𝑧 ∈ V
5	4	brbigcup 36640	. . 3 ⊢ (𝑦 Bigcup 𝑧 ↔ ∪ 𝑦 = 𝑧)
6		vex 3455	. . . 4 ⊢ 𝑦 ∈ V
7		eleq1w 2844	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V))
8		unieq 4878	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ∪ 𝑥 = ∪ 𝑦)
9	8	eqeq2d 2772	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑡 = ∪ 𝑥 ↔ 𝑡 = ∪ 𝑦))
10	7, 9	anbi12d 644	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦)))
11	6	biantrur 540	. . . . 5 ⊢ (𝑡 = ∪ 𝑦 ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦))
12	10, 11	bitr4di 292	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ 𝑡 = ∪ 𝑦))
13		eqeq1 2765	. . . 4 ⊢ (𝑡 = 𝑧 → (𝑡 = ∪ 𝑦 ↔ 𝑧 = ∪ 𝑦))
14		df-mpt 5187	. . . 4 ⊢ (𝑥 ∈ V ↦ ∪ 𝑥) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥)}
15	6, 4, 12, 13, 14	brab 5518	. . 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 5767	1 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ∪ cuni 4867   class class class wbr 5103   ↦ cmpt 5186   Bigcup cbigcup 36576

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-symdif 4199  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7999  df-2nd 8000  df-txp 36596  df-bigcup 36600

This theorem is used by:  fobigcup  36642