Theorem fo1st 6381

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 2824	. . . . . 6 ⊢ 𝑥 ∈ V
2	1	snex 4317	. . . . 5 ⊢ {𝑥} ∈ V
3	2	dmex 5044	. . . 4 ⊢ dom {𝑥} ∈ V
4	3	uniex 4578	. . 3 ⊢ ∪ dom {𝑥} ∈ V
5		df-1st 6364	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
6	4, 5	fnmpti 5507	. 2 ⊢ 1st Fn V
7	5	rnmpt 5025	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
8		vex 2824	. . . . 5 ⊢ 𝑦 ∈ V
9	8, 8	opex 4364	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
10	8, 8	op1sta 5264	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
11	10	eqcomi 2242	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
12		sneq 3716	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
13	12	dmeqd 4978	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
14	13	unieqd 3941	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
15	14	eqeq2d 2250	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ∪ dom {𝑥} ↔ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}))
16	15	rspcev 2929	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
17	9, 11, 16	mp2an 430	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
18	8, 17	2th 174	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
19	18	abbi2i 2353	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
20	7, 19	eqtr4i 2262	. 2 ⊢ ran 1st = V
21		df-fo 5378	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
22	6, 20, 21	mpbir2an 955	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1402   ∈ wcel 2209  {cab 2224  ∃wrex 2529  Vcvv 2821  {csn 3705  ⟨cop 3708  ∪ cuni 3930  dom cdm 4769  ran crn 4770   Fn wfn 5367  –onto→wfo 5370  1^st c1st 6362

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-fun 5374  df-fn 5375  df-fo 5378  df-1st 6364

This theorem is referenced by:  1stcof  6387  1stexg  6391  df1st2  6445  1stconst  6447  algrflem  6455  algrflemg  6456  suplocexprlemell  8070  suplocexprlem2b  8071  suplocexprlemlub  8081  upxp  15296  uptx  15298  cnmpt1st  15312