Theorem fo1st 7985

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 5389	. . . . 5 ⊢ {𝑥} ∈ V
2	1	dmex 7885	. . . 4 ⊢ dom {𝑥} ∈ V
3	2	uniex 7719	. . 3 ⊢ ∪ dom {𝑥} ∈ V
4		df-1st 7965	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5	3, 4	fnmpti 6659	. 2 ⊢ 1st Fn V
6	4	rnmpt 5929	. . 3 ⊢ ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
7		vex 3457	. . . . 5 ⊢ 𝑦 ∈ V
8		opex 5428	. . . . . 6 ⊢ ⟨𝑦, 𝑦⟩ ∈ V
9	7, 7	op1sta 6207	. . . . . . 7 ⊢ ∪ dom {⟨𝑦, 𝑦⟩} = 𝑦
10	9	eqcomi 2770	. . . . . 6 ⊢ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}
11		sneq 4589	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
12	11	dmeqd 5877	. . . . . . . 8 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
13	12	unieqd 4875	. . . . . . 7 ⊢ (𝑥 = ⟨𝑦, 𝑦⟩ → ∪ dom {𝑥} = ∪ dom {⟨𝑦, 𝑦⟩})
14	13	rspceeqv 3603	. . . . . 6 ⊢ ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = ∪ dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
15	8, 10, 14	mp2an 702	. . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}
16	7, 15	2th 266	. . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥})
17	16	eqabi 2896	. . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ dom {𝑥}}
18	6, 17	eqtr4i 2787	. 2 ⊢ ran 1st = V
19		df-fo 6522	. 2 ⊢ (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
20	5, 18, 19	mpbir2an 721	1 ⊢ 1st :V–onto→V

Syntax hints:   = wceq 1559   ∈ wcel 2141  {cab 2739  ∃wrex 3085  Vcvv 3453  {csn 4579  ⟨cop 4585  ∪ cuni 4862  dom cdm 5643  ran crn 5644   Fn wfn 6511  –onto→wfo 6514  1^st c1st 7963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-fun 6518  df-fn 6519  df-fo 6522  df-1st 7965

This theorem is referenced by:  br1steqg  7987  1stcof  7995  df1st2  8071  1stconst  8073  fsplit  8090  opco1  8096  fpwwe  10598  axpre-sup  11121  homadm  18064  homacd  18065  dmaf  18073  cdaf  18074  1stf1  18215  1stf2  18216  1stfcl  18220  upxp  23671  uptx  23673  cnmpt1st  23716  bcthlem4  25377  uniiccdif  25628  precsexlem10  28297  precsexlem11  28298  vafval  30763  smfval  30765  0vfval  30766  vsfval  30793  xppreima  32808  xppreima2  32814  1stpreimas  32869  1stpreima  32870  fsuppcurry2  32888  gsummpt2d  33190  cnre2csqima  34169  poimirlem26  38106  poimirlem27  38107