Theorem fo1st 7230

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		snex 4938	. . . . 5 ⊢ {𝑥} ∈ V
2	1	dmex 7141	. . . 4 ⊢ dom {𝑥} ∈ V
3	2	uniex 6995	. . 3 ⊢ ∪ dom {𝑥} ∈ V
4		df-1st 7210	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5	3, 4	fnmpti 6060	. 2 ⊢ 1st Fn V
6	4	rnmpt 5403	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
7		vex 3234	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 4962	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op1sta 5654	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2660	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
11		sneq 4220	. . . . . . . . . 10 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	dmeqd 5358	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
13	12	unieqd 4478	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
14	13	eqeq2d 2661	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = ∪ dom {𝑥} ↔ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}))
15	14	rspcev 3340	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
16	8, 10, 15	mp2an 708	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
17	7, 16	2th 254	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
18	17	abbi2i 2767	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
19	6, 18	eqtr4i 2676	. 2 ⊢ ran 1st = V
20		df-fo 5932	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
21	5, 19, 20	mpbir2an 975	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1523   ∈ wcel 2030  {cab 2637  ∃wrex 2942  Vcvv 3231  {csn 4210  ⟨cop 4216  ∪ cuni 4468  dom cdm 5143  ran crn 5144   Fn wfn 5921  –onto→wfo 5924  1^st c1st 7208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-fo 5932  df-1st 7210

This theorem is referenced by:  br1steqg  7232  1stcof  7240  df1st2  7308  1stconst  7310  fsplit  7327  algrflem  7331  fpwwe  9506  axpre-sup  10028  homadm  16737  homacd  16738  dmaf  16746  cdaf  16747  1stf1  16879  1stf2  16880  1stfcl  16884  upxp  21474  uptx  21476  cnmpt1st  21519  bcthlem4  23170  uniiccdif  23392  vafval  27586  smfval  27588  0vfval  27589  vsfval  27616  xppreima  29577  xppreima2  29578  1stpreimas  29611  1stpreima  29612  gsummpt2d  29909  cnre2csqima  30085  poimirlem26  33565  poimirlem27  33566