Theorem fo1st 6063

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 2692	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snex 4117	. . . . 5 ⊢ {𝑥} ∈ V
3	2	dmex 4813	. . . 4 ⊢ dom {𝑥} ∈ V
4	3	uniex 4367	. . 3 ⊢ ∪ dom {𝑥} ∈ V
5		df-1st 6046	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
6	4, 5	fnmpti 5259	. 2 ⊢ 1st Fn V
7	5	rnmpt 4795	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
8		vex 2692	. . . . 5 ⊢ 𝑦 ∈ V
9	8, 8	opex 4159	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
10	8, 8	op1sta 5028	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
11	10	eqcomi 2144	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
12		sneq 3543	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
13	12	dmeqd 4749	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
14	13	unieqd 3755	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
15	14	eqeq2d 2152	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ∪ dom {𝑥} ↔ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}))
16	15	rspcev 2793	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
17	9, 11, 16	mp2an 423	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
18	8, 17	2th 173	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
19	18	abbi2i 2255	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
20	7, 19	eqtr4i 2164	. 2 ⊢ ran 1st = V
21		df-fo 5137	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
22	6, 20, 21	mpbir2an 927	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1332   ∈ wcel 1481  {cab 2126  ∃wrex 2418  Vcvv 2689  {csn 3532  ⟨cop 3535  ∪ cuni 3744  dom cdm 4547  ran crn 4548   Fn wfn 5126  –onto→wfo 5129  1^st c1st 6044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-fun 5133  df-fn 5134  df-fo 5137  df-1st 6046

This theorem is referenced by:  1stcof  6069  1stexg  6073  df1st2  6124  1stconst  6126  algrflem  6134  algrflemg  6135  suplocexprlemell  7545  suplocexprlem2b  7546  suplocexprlemlub  7556  upxp  12480  uptx  12482  cnmpt1st  12496