Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1o00 Structured version   Visualization version   GIF version

Theorem f1o00 6138
 Description: One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
Assertion
Ref Expression
f1o00 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

Proof of Theorem f1o00
StepHypRef Expression
1 dff1o4 6112 . 2 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2 fn0 5978 . . . . . 6 (𝐹 Fn ∅ ↔ 𝐹 = ∅)
32biimpi 206 . . . . 5 (𝐹 Fn ∅ → 𝐹 = ∅)
43adantr 481 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐹 = ∅)
5 dm0 5309 . . . . 5 dom ∅ = ∅
6 cnveq 5266 . . . . . . . . . 10 (𝐹 = ∅ → 𝐹 = ∅)
7 cnv0 5504 . . . . . . . . . 10 ∅ = ∅
86, 7syl6eq 2671 . . . . . . . . 9 (𝐹 = ∅ → 𝐹 = ∅)
92, 8sylbi 207 . . . . . . . 8 (𝐹 Fn ∅ → 𝐹 = ∅)
109fneq1d 5949 . . . . . . 7 (𝐹 Fn ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
1110biimpa 501 . . . . . 6 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → ∅ Fn 𝐴)
12 fndm 5958 . . . . . 6 (∅ Fn 𝐴 → dom ∅ = 𝐴)
1311, 12syl 17 . . . . 5 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → dom ∅ = 𝐴)
145, 13syl5reqr 2670 . . . 4 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → 𝐴 = ∅)
154, 14jca 554 . . 3 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) → (𝐹 = ∅ ∧ 𝐴 = ∅))
162biimpri 218 . . . . 5 (𝐹 = ∅ → 𝐹 Fn ∅)
1716adantr 481 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn ∅)
18 eqid 2621 . . . . . 6 ∅ = ∅
19 fn0 5978 . . . . . 6 (∅ Fn ∅ ↔ ∅ = ∅)
2018, 19mpbir 221 . . . . 5 ∅ Fn ∅
218fneq1d 5949 . . . . . 6 (𝐹 = ∅ → (𝐹 Fn 𝐴 ↔ ∅ Fn 𝐴))
22 fneq2 5948 . . . . . 6 (𝐴 = ∅ → (∅ Fn 𝐴 ↔ ∅ Fn ∅))
2321, 22sylan9bb 735 . . . . 5 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn 𝐴 ↔ ∅ Fn ∅))
2420, 23mpbiri 248 . . . 4 ((𝐹 = ∅ ∧ 𝐴 = ∅) → 𝐹 Fn 𝐴)
2517, 24jca 554 . . 3 ((𝐹 = ∅ ∧ 𝐴 = ∅) → (𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴))
2615, 25impbii 199 . 2 ((𝐹 Fn ∅ ∧ 𝐹 Fn 𝐴) ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
271, 26bitri 264 1 (𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480  ∅c0 3897  ◡ccnv 5083  dom cdm 5084   Fn wfn 5852  –1-1-onto→wf1o 5856 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864 This theorem is referenced by:  fo00  6139  f1o0  6140  en0  7979  symgbas0  17754  derang0  30912  poimirlem28  33108
 Copyright terms: Public domain W3C validator