Theorem fo1st 6301

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 2802	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snex 4268	. . . . 5 ⊢ {𝑥} ∈ V
3	2	dmex 4990	. . . 4 ⊢ dom {𝑥} ∈ V
4	3	uniex 4527	. . 3 ⊢ ∪ dom {𝑥} ∈ V
5		df-1st 6284	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
6	4, 5	fnmpti 5451	. 2 ⊢ 1st Fn V
7	5	rnmpt 4971	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
8		vex 2802	. . . . 5 ⊢ 𝑦 ∈ V
9	8, 8	opex 4314	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
10	8, 8	op1sta 5209	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
11	10	eqcomi 2233	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
12		sneq 3677	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
13	12	dmeqd 4924	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
14	13	unieqd 3898	. . . . . . . 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 2907	. . . . . 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 5323	. 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 2799  {csn 3666  ⟨cop 3669  ∪ cuni 3887  dom cdm 4718  ran crn 4719   Fn wfn 5312  –onto→wfo 5315  1^st c1st 6282

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 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-fun 5319  df-fn 5320  df-fo 5323  df-1st 6284

This theorem is referenced by:  1stcof  6307  1stexg  6311  df1st2  6363  1stconst  6365  algrflem  6373  algrflemg  6374  suplocexprlemell  7896  suplocexprlem2b  7897  suplocexprlemlub  7907  upxp  14940  uptx  14942  cnmpt1st  14956