Theorem fobigcup 31177

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 6853	. . . 4 ⊢ (𝑥 ∈ V → ∪ 𝑥 ∈ V)
2	1	rgen 2906	. . 3 ⊢ ∀𝑥 ∈ V ∪ 𝑥 ∈ V
3		dfbigcup2 31176	. . . 4 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥)
4	3	mptfng 5932	. . 3 ⊢ (∀𝑥 ∈ V ∪ 𝑥 ∈ V ↔ Bigcup Fn V)
5	2, 4	mpbi 219	. 2 ⊢ Bigcup Fn V
6	3	rnmpt 5292	. . 3 ⊢ ran Bigcup = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥}
7		vex 3176	. . . . 5 ⊢ 𝑦 ∈ V
8		snex 4835	. . . . . 6 ⊢ {𝑦} ∈ V
9	7	unisn 4387	. . . . . . 7 ⊢ ∪ {𝑦} = 𝑦
10	9	eqcomi 2619	. . . . . 6 ⊢ 𝑦 = ∪ {𝑦}
11		unieq 4380	. . . . . . . 8 ⊢ (𝑥 = {𝑦} → ∪ 𝑥 = ∪ {𝑦})
12	11	eqeq2d 2620	. . . . . . 7 ⊢ (𝑥 = {𝑦} → (𝑦 = ∪ 𝑥 ↔ 𝑦 = ∪ {𝑦}))
13	12	rspcev 3282	. . . . . 6 ⊢ (({𝑦} ∈ V ∧ 𝑦 = ∪ {𝑦}) → ∃𝑥 ∈ V 𝑦 = ∪ 𝑥)
14	8, 10, 13	mp2an 704	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥
15	7, 14	2th 253	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥)
16	15	abbi2i 2725	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥}
17	6, 16	eqtr4i 2635	. 2 ⊢ ran Bigcup = V
18		df-fo 5810	. 2 ⊢ ( Bigcup :V–onto→V ↔ ( Bigcup Fn V ∧ ran Bigcup = V))
19	5, 17, 18	mpbir2an 957	1 ⊢ Bigcup :V–onto→V

Syntax hints:   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173  {csn 4125  ∪ cuni 4372  ran crn 5039   Fn wfn 5799  –onto→wfo 5802   Bigcup cbigcup 31110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-symdif 3806  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-eprel 4949  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060  df-txp 31130  df-bigcup 31134

This theorem is referenced by:  fnbigcup  31178