Theorem fobigcup 33356

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 7460	. . . 4 ⊢ (𝑥 ∈ V → ∪ 𝑥 ∈ V)
2	1	rgen 3148	. . 3 ⊢ ∀𝑥 ∈ V ∪ 𝑥 ∈ V
3		dfbigcup2 33355	. . . 4 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)
4	3	mptfng 6482	. . 3 ⊢ (∀𝑥 ∈ V ∪ 𝑥 ∈ V ↔ Bigcup Fn V)
5	2, 4	mpbi 232	. 2 ⊢ Bigcup Fn V
6	3	rnmpt 5822	. . 3 ⊢ ran Bigcup = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥}
7		vex 3498	. . . . 5 ⊢ 𝑦 ∈ V
8		snex 5324	. . . . . 6 ⊢ {𝑦} ∈ V
9	7	unisn 4848	. . . . . . 7 ⊢ ∪ {𝑦} = 𝑦
10	9	eqcomi 2830	. . . . . 6 ⊢ 𝑦 = ∪ {𝑦}
11		unieq 4840	. . . . . . 7 ⊢ (𝑥 = {𝑦} → ∪ 𝑥 = ∪ {𝑦})
12	11	rspceeqv 3638	. . . . . 6 ⊢ (({𝑦} ∈ V ∧ 𝑦 = ∪ {𝑦}) → ∃𝑥 ∈ V 𝑦 = ∪ 𝑥)
13	8, 10, 12	mp2an 690	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥
14	7, 13	2th 266	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥)
15	14	abbi2i 2953	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥}
16	6, 15	eqtr4i 2847	. 2 ⊢ ran Bigcup = V
17		df-fo 6356	. 2 ⊢ ( Bigcup :V–onto→V ↔ ( Bigcup Fn V ∧ ran Bigcup = V))
18	5, 16, 17	mpbir2an 709	1 ⊢ Bigcup :V–onto→V

Syntax hints:   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  ∃wrex 3139  Vcvv 3495  {csn 4561  ∪ cuni 4832  ran crn 5551   Fn wfn 6345  –onto→wfo 6348   Bigcup cbigcup 33290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-symdif 4219  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-eprel 5460  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fo 6356  df-fv 6358  df-1st 7683  df-2nd 7684  df-txp 33310  df-bigcup 33314

This theorem is referenced by:  fnbigcup  33357