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Theorem fo1st 5811
Description: The 1st 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.
StepHypRef Expression
1 vex 2577 . . . . . 6 𝑥 ∈ V
2 snexgOLD 3962 . . . . . 6 (𝑥 ∈ V → {𝑥} ∈ V)
31, 2ax-mp 7 . . . . 5 {𝑥} ∈ V
43dmex 4625 . . . 4 dom {𝑥} ∈ V
54uniex 4201 . . 3 dom {𝑥} ∈ V
6 df-1st 5794 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
75, 6fnmpti 5054 . 2 1st Fn V
86rnmpt 4609 . . 3 ran 1st = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = dom {𝑥}}
9 vex 2577 . . . . 5 𝑦 ∈ V
109, 9opex 3993 . . . . . 6 𝑦, 𝑦⟩ ∈ V
119, 9op1sta 4829 . . . . . . 7 dom {⟨𝑦, 𝑦⟩} = 𝑦
1211eqcomi 2060 . . . . . 6 𝑦 = dom {⟨𝑦, 𝑦⟩}
13 sneq 3413 . . . . . . . . . 10 (𝑥 = ⟨𝑦, 𝑦⟩ → {𝑥} = {⟨𝑦, 𝑦⟩})
1413dmeqd 4564 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
1514unieqd 3618 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑦⟩ → dom {𝑥} = dom {⟨𝑦, 𝑦⟩})
1615eqeq2d 2067 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑦⟩ → (𝑦 = dom {𝑥} ↔ 𝑦 = dom {⟨𝑦, 𝑦⟩}))
1716rspcev 2673 . . . . . 6 ((⟨𝑦, 𝑦⟩ ∈ V ∧ 𝑦 = dom {⟨𝑦, 𝑦⟩}) → ∃𝑥 ∈ V 𝑦 = dom {𝑥})
1810, 12, 17mp2an 410 . . . . 5 𝑥 ∈ V 𝑦 = dom {𝑥}
199, 182th 167 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = dom {𝑥})
2019abbi2i 2168 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = dom {𝑥}}
218, 20eqtr4i 2079 . 2 ran 1st = V
22 df-fo 4935 . 2 (1st :V–onto→V ↔ (1st Fn V ∧ ran 1st = V))
237, 21, 22mpbir2an 860 1 1st :V–onto→V
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  {cab 2042  wrex 2324  Vcvv 2574  {csn 3402  cop 3405   cuni 3607  dom cdm 4372  ran crn 4373   Fn wfn 4924  ontowfo 4927  1st c1st 5792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-fun 4931  df-fn 4932  df-fo 4935  df-1st 5794
This theorem is referenced by:  1stcof  5817  1stexg  5821  df1st2  5867  1stconst  5869  algrflem  5877  algrflemg  5878
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