Theorem fo1st 7386

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 5064	. . . . 5 ⊢ {𝑥} ∈ V
2	1	dmex 7297	. . . 4 ⊢ dom {𝑥} ∈ V
3	2	uniex 7151	. . 3 ⊢ ∪ dom {𝑥} ∈ V
4		df-1st 7366	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5	3, 4	fnmpti 6200	. 2 ⊢ 1st Fn V
6	4	rnmpt 5540	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
7		vex 3353	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5088	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op1sta 5802	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2774	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
11		sneq 4344	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	dmeqd 5494	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
13	12	unieqd 4604	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3479	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
15	8, 10, 14	mp2an 683	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
16	7, 15	2th 255	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
17	16	abbi2i 2881	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
18	6, 17	eqtr4i 2790	. 2 ⊢ ran 1st = V
19		df-fo 6074	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
20	5, 18, 19	mpbir2an 702	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1652   ∈ wcel 2155  {cab 2751  ∃wrex 3056  Vcvv 3350  {csn 4334  ⟨cop 4340  ∪ cuni 4594  dom cdm 5277  ran crn 5278   Fn wfn 6063  –onto→wfo 6066  1^st c1st 7364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-fun 6070  df-fn 6071  df-fo 6074  df-1st 7366

This theorem is referenced by:  br1steqg  7388  1stcof  7396  df1st2  7465  1stconst  7467  fsplit  7484  algrflem  7488  fpwwe  9721  axpre-sup  10243  homadm  16955  homacd  16956  dmaf  16964  cdaf  16965  1stf1  17098  1stf2  17099  1stfcl  17103  upxp  21706  uptx  21708  cnmpt1st  21751  bcthlem4  23404  uniiccdif  23636  vafval  27914  smfval  27916  0vfval  27917  vsfval  27944  xppreima  29899  xppreima2  29900  1stpreimas  29932  1stpreima  29933  gsummpt2d  30228  cnre2csqima  30404  poimirlem26  33859  poimirlem27  33860