Theorem fo1st 6351

Description: The 1^st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
fo1st	⊢ 1st :V–onto→V

Proof of Theorem fo1st

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

Step	Hyp	Ref	Expression
1		vex 2816	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snex 4298	. . . . 5 ⊢ {𝑥} ∈ V
3	2	dmex 5024	. . . 4 ⊢ dom {𝑥} ∈ V
4	3	uniex 4558	. . 3 ⊢ ∪ dom {𝑥} ∈ V
5		df-1st 6334	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
6	4, 5	fnmpti 5487	. 2 ⊢ 1st Fn V
7	5	rnmpt 5005	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
8		vex 2816	. . . . 5 ⊢ 𝑦 ∈ V
9	8, 8	opex 4345	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
10	8, 8	op1sta 5244	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
11	10	eqcomi 2236	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
12		sneq 3700	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
13	12	dmeqd 4958	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
14	13	unieqd 3925	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
15	14	eqeq2d 2244	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ∪ dom {𝑥} ↔ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}))
16	15	rspcev 2921	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
17	9, 11, 16	mp2an 426	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
18	8, 17	2th 174	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
19	18	abbi2i 2347	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
20	7, 19	eqtr4i 2256	. 2 ⊢ ran 1st = V
21		df-fo 5358	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
22	6, 20, 21	mpbir2an 951	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1398   ∈ wcel 2203  {cab 2218  ∃wrex 2521  Vcvv 2813  {csn 3689  ⟨cop 3692  ∪ cuni 3914  dom cdm 4749  ran crn 4750   Fn wfn 5347  –onto→wfo 5350  1^st c1st 6332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-fun 5354  df-fn 5355  df-fo 5358  df-1st 6334

This theorem is referenced by:  1stcof  6357  1stexg  6361  df1st2  6415  1stconst  6417  algrflem  6425  algrflemg  6426  suplocexprlemell  8028  suplocexprlem2b  8029  suplocexprlemlub  8039  upxp  15137  uptx  15139  cnmpt1st  15153