Theorem fobigcup 36209

Description: Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)

Assertion

Ref	Expression
fobigcup	⊢ Bigcup :V–onto→V

Proof of Theorem fobigcup

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniexg 7718	. . . 4 ⊢ (𝑥 ∈ V → ∪ 𝑥 ∈ V)
2	1	rgen 3077	. . 3 ⊢ ∀𝑥 ∈ V ∪ 𝑥 ∈ V
3		dfbigcup2 36208	. . . 4 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)
4	3	mptfng 6655	. . 3 ⊢ (∀𝑥 ∈ V ∪ 𝑥 ∈ V ↔ Bigcup Fn V)
5	2, 4	mpbi 232	. 2 ⊢ Bigcup Fn V
6	3	rnmpt 5929	. . 3 ⊢ ran Bigcup = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥}
7		vex 3457	. . . . 5 ⊢ 𝑦 ∈ V
8		vsnex 5389	. . . . . 6 ⊢ {𝑦} ∈ V
9		unisnv 4882	. . . . . . 7 ⊢ ∪ {𝑦} = 𝑦
10	9	eqcomi 2770	. . . . . 6 ⊢ 𝑦 = ∪ {𝑦}
11		unieq 4873	. . . . . . 7 ⊢ (𝑥 = {𝑦} → ∪ 𝑥 = ∪ {𝑦})
12	11	rspceeqv 3603	. . . . . 6 ⊢ (({𝑦} ∈ V ∧ 𝑦 = ∪ {𝑦}) → ∃𝑥 ∈ V 𝑦 = ∪ 𝑥)
13	8, 10, 12	mp2an 702	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥
14	7, 13	2th 266	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥)
15	14	eqabi 2896	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥}
16	6, 15	eqtr4i 2787	. 2 ⊢ ran Bigcup = V
17		df-fo 6522	. 2 ⊢ ( Bigcup :V–onto→V ↔ ( Bigcup Fn V ∧ ran Bigcup = V))
18	5, 16, 17	mpbir2an 721	1 ⊢ Bigcup :V–onto→V

Syntax hints:   = wceq 1559   ∈ wcel 2141  {cab 2739  ∀wral 3075  ∃wrex 3085  Vcvv 3453  {csn 4579  ∪ cuni 4862  ran crn 5644   Fn wfn 6511  –onto→wfo 6514   Bigcup cbigcup 36143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-symdif 4203  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-eprel 5543  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-fo 6522  df-fv 6524  df-1st 7965  df-2nd 7966  df-txp 36163  df-bigcup 36167

This theorem is referenced by:  fnbigcup  36210