Theorem fo1st 8002

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		vsnex 5406	. . . . 5 ⊢ {𝑥} ∈ V
2	1	dmex 7902	. . . 4 ⊢ dom {𝑥} ∈ V
3	2	uniex 7739	. . 3 ⊢ ∪ dom {𝑥} ∈ V
4		df-1st 7982	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5	3, 4	fnmpti 6678	. 2 ⊢ 1st Fn V
6	4	rnmpt 5947	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
7		vex 3459	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5445	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op1sta 6226	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2772	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
11		sneq 4599	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	dmeqd 5895	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
13	12	unieqd 4885	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3604	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
15	8, 10, 14	mp2an 704	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
16	7, 15	2th 267	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
17	16	eqabi 2898	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
18	6, 17	eqtr4i 2789	. 2 ⊢ ran 1st = V
19		df-fo 6542	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
20	5, 18, 19	mpbir2an 723	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1570   ∈ wcel 2143  {cab 2741  ∃wrex 3089  Vcvv 3455  {csn 4589  ⟨cop 4595  ∪ cuni 4872  dom cdm 5661  ran crn 5662   Fn wfn 6531  –onto→wfo 6534  1^st c1st 7980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-fun 6538  df-fn 6539  df-fo 6542  df-1st 7982

This theorem is referenced by:  br1steqg  8004  1stcof  8012  df1st2  8089  1stconst  8091  fsplit  8108  opco1  8114  fpwwe  10626  axpre-sup  11149  homadm  18092  homacd  18093  dmaf  18101  cdaf  18102  1stf1  18243  1stf2  18244  1stfcl  18248  upxp  23780  uptx  23782  cnmpt1st  23825  bcthlem4  25486  uniiccdif  25737  precsexlem10  28409  precsexlem11  28410  vafval  30955  smfval  30957  0vfval  30958  vsfval  30985  xppreima  32990  xppreima2  32996  1stpreimas  33051  1stpreima  33052  fsuppcurry2  33070  gsummpt2d  33369  cnre2csqima  34301  poimirlem26  38297  poimirlem27  38298