Theorem fo1st 7991

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 5392	. . . . 5 ⊢ {𝑥} ∈ V
2	1	dmex 7888	. . . 4 ⊢ dom {𝑥} ∈ V
3	2	uniex 7720	. . 3 ⊢ ∪ dom {𝑥} ∈ V
4		df-1st 7971	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5	3, 4	fnmpti 6664	. 2 ⊢ 1st Fn V
6	4	rnmpt 5924	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
7		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5427	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op1sta 6201	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2739	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
11		sneq 4602	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	dmeqd 5872	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
13	12	unieqd 4887	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3614	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
15	8, 10, 14	mp2an 692	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
16	7, 15	2th 264	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
17	16	eqabi 2864	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
18	6, 17	eqtr4i 2756	. 2 ⊢ ran 1st = V
19		df-fo 6520	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
20	5, 18, 19	mpbir2an 711	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1540   ∈ wcel 2109  {cab 2708  ∃wrex 3054  Vcvv 3450  {csn 4592  ⟨cop 4598  ∪ cuni 4874  dom cdm 5641  ran crn 5642   Fn wfn 6509  –onto→wfo 6512  1^st c1st 7969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-fun 6516  df-fn 6517  df-fo 6520  df-1st 7971

This theorem is referenced by:  br1steqg  7993  1stcof  8001  df1st2  8080  1stconst  8082  fsplit  8099  opco1  8105  fpwwe  10606  axpre-sup  11129  homadm  18009  homacd  18010  dmaf  18018  cdaf  18019  1stf1  18160  1stf2  18161  1stfcl  18165  upxp  23517  uptx  23519  cnmpt1st  23562  bcthlem4  25234  uniiccdif  25486  precsexlem10  28125  precsexlem11  28126  vafval  30539  smfval  30541  0vfval  30542  vsfval  30569  xppreima  32576  xppreima2  32582  1stpreimas  32636  1stpreima  32637  fsuppcurry2  32656  gsummpt2d  32996  cnre2csqima  33908  poimirlem26  37647  poimirlem27  37648