Theorem fobigcup 33365

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 7469	. . . 4 ⊢ (𝑥 ∈ V → ∪ 𝑥 ∈ V)
2	1	rgen 3151	. . 3 ⊢ ∀𝑥 ∈ V ∪ 𝑥 ∈ V
3		dfbigcup2 33364	. . . 4 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)
4	3	mptfng 6490	. . 3 ⊢ (∀𝑥 ∈ V ∪ 𝑥 ∈ V ↔ Bigcup Fn V)
5	2, 4	mpbi 232	. 2 ⊢ Bigcup Fn V
6	3	rnmpt 5830	. . 3 ⊢ ran Bigcup = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥}
7		vex 3500	. . . . 5 ⊢ 𝑦 ∈ V
8		snex 5335	. . . . . 6 ⊢ {𝑦} ∈ V
9	7	unisn 4861	. . . . . . 7 ⊢ ∪ {𝑦} = 𝑦
10	9	eqcomi 2833	. . . . . 6 ⊢ 𝑦 = ∪ {𝑦}
11		unieq 4852	. . . . . . 7 ⊢ (𝑥 = {𝑦} → ∪ 𝑥 = ∪ {𝑦})
12	11	rspceeqv 3641	. . . . . 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 2956	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥}
16	6, 15	eqtr4i 2850	. 2 ⊢ ran Bigcup = V
17		df-fo 6364	. 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 1536   ∈ wcel 2113  {cab 2802  ∀wral 3141  ∃wrex 3142  Vcvv 3497  {csn 4570  ∪ cuni 4841  ran crn 5559   Fn wfn 6353  –onto→wfo 6356   Bigcup cbigcup 33299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-symdif 4222  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-eprel 5468  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fo 6364  df-fv 6366  df-1st 7692  df-2nd 7693  df-txp 33319  df-bigcup 33323

This theorem is referenced by:  fnbigcup  33366