Theorem fo1st 7824

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 5349	. . . . 5 ⊢ {𝑥} ∈ V
2	1	dmex 7732	. . . 4 ⊢ dom {𝑥} ∈ V
3	2	uniex 7572	. . 3 ⊢ ∪ dom {𝑥} ∈ V
4		df-1st 7804	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5	3, 4	fnmpti 6560	. 2 ⊢ 1st Fn V
6	4	rnmpt 5853	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
7		vex 3426	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5373	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op1sta 6117	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2747	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
11		sneq 4568	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	dmeqd 5803	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
13	12	unieqd 4850	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3567	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
15	8, 10, 14	mp2an 688	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
16	7, 15	2th 263	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
17	16	abbi2i 2878	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
18	6, 17	eqtr4i 2769	. 2 ⊢ ran 1st = V
19		df-fo 6424	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
20	5, 18, 19	mpbir2an 707	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064  Vcvv 3422  {csn 4558  ⟨cop 4564  ∪ cuni 4836  dom cdm 5580  ran crn 5581   Fn wfn 6413  –onto→wfo 6416  1^st c1st 7802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-fun 6420  df-fn 6421  df-fo 6424  df-1st 7804

This theorem is referenced by:  br1steqg  7826  1stcof  7834  df1st2  7909  1stconst  7911  fsplit  7928  fsplitOLD  7929  opco1  7935  fpwwe  10333  axpre-sup  10856  homadm  17671  homacd  17672  dmaf  17680  cdaf  17681  1stf1  17825  1stf2  17826  1stfcl  17830  upxp  22682  uptx  22684  cnmpt1st  22727  bcthlem4  24396  uniiccdif  24647  vafval  28866  smfval  28868  0vfval  28869  vsfval  28896  xppreima  30884  xppreima2  30889  1stpreimas  30940  1stpreima  30941  fsuppcurry2  30963  gsummpt2d  31211  cnre2csqima  31763  poimirlem26  35730  poimirlem27  35731