Theorem fobigcup 36285

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 7735	. . . 4 ⊢ (𝑥 ∈ V → ∪ 𝑥 ∈ V)
2	1	rgen 3087	. . 3 ⊢ ∀𝑥 ∈ V ∪ 𝑥 ∈ V
3		dfbigcup2 36284	. . . 4 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)
4	3	mptfng 6672	. . 3 ⊢ (∀𝑥 ∈ V ∪ 𝑥 ∈ V ↔ Bigcup Fn V)
5	2, 4	mpbi 233	. 2 ⊢ Bigcup Fn V
6	3	rnmpt 5945	. . 3 ⊢ ran Bigcup = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥}
7		vex 3467	. . . . 5 ⊢ 𝑦 ∈ V
8		vsnex 5404	. . . . . 6 ⊢ {𝑦} ∈ V
9		unisnv 4893	. . . . . . 7 ⊢ ∪ {𝑦} = 𝑦
10	9	eqcomi 2778	. . . . . 6 ⊢ 𝑦 = ∪ {𝑦}
11		unieq 4884	. . . . . . 7 ⊢ (𝑥 = {𝑦} → ∪ 𝑥 = ∪ {𝑦})
12	11	rspceeqv 3613	. . . . . 6 ⊢ (({𝑦} ∈ V ∧ 𝑦 = ∪ {𝑦}) → ∃𝑥 ∈ V 𝑦 = ∪ 𝑥)
13	8, 10, 12	mp2an 704	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥
14	7, 13	2th 267	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥)
15	14	eqabi 2904	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥}
16	6, 15	eqtr4i 2795	. 2 ⊢ ran Bigcup = V
17		df-fo 6540	. 2 ⊢ ( Bigcup :V–onto→V ↔ ( Bigcup Fn V ∧ ran Bigcup = V))
18	5, 16, 17	mpbir2an 723	1 ⊢ Bigcup :V–onto→V

Syntax hints:   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095  Vcvv 3463  {csn 4591  ∪ cuni 4873  ran crn 5660   Fn wfn 6529  –onto→wfo 6532   Bigcup cbigcup 36219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-symdif 4214  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-eprel 5559  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-fo 6540  df-fv 6542  df-1st 7982  df-2nd 7983  df-txp 36239  df-bigcup 36243

This theorem is referenced by:  fnbigcup  36286