Theorem fo1st 6315

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 2803	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snex 4273	. . . . 5 ⊢ {𝑥} ∈ V
3	2	dmex 4997	. . . 4 ⊢ dom {𝑥} ∈ V
4	3	uniex 4532	. . 3 ⊢ ∪ dom {𝑥} ∈ V
5		df-1st 6298	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
6	4, 5	fnmpti 5458	. 2 ⊢ 1st Fn V
7	5	rnmpt 4978	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
8		vex 2803	. . . . 5 ⊢ 𝑦 ∈ V
9	8, 8	opex 4319	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
10	8, 8	op1sta 5216	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
11	10	eqcomi 2233	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
12		sneq 3678	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
13	12	dmeqd 4931	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
14	13	unieqd 3902	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
15	14	eqeq2d 2241	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ∪ dom {𝑥} ↔ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}))
16	15	rspcev 2908	. . . . . 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 2344	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
20	7, 19	eqtr4i 2253	. 2 ⊢ ran 1st = V
21		df-fo 5330	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
22	6, 20, 21	mpbir2an 948	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1395   ∈ wcel 2200  {cab 2215  ∃wrex 2509  Vcvv 2800  {csn 3667  ⟨cop 3670  ∪ cuni 3891  dom cdm 4723  ran crn 4724   Fn wfn 5319  –onto→wfo 5322  1^st c1st 6296

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-fun 5326  df-fn 5327  df-fo 5330  df-1st 6298

This theorem is referenced by:  1stcof  6321  1stexg  6325  df1st2  6379  1stconst  6381  algrflem  6389  algrflemg  6390  suplocexprlemell  7923  suplocexprlem2b  7924  suplocexprlemlub  7934  upxp  14986  uptx  14988  cnmpt1st  15002