MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f11o Structured version   Visualization version   GIF version

Theorem f11o 7970
Description: Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
Hypothesis
Ref Expression
f11o.1 𝐹 ∈ V
Assertion
Ref Expression
f11o (𝐹:𝐴1-1𝐵 ↔ ∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴   𝑥,𝐵

Proof of Theorem f11o
StepHypRef Expression
1 f11o.1 . . . 4 𝐹 ∈ V
21ffoss 7969 . . 3 (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
32anbi1i 624 . 2 ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ↔ (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
4 df-f1 6568 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
5 dff1o3 6855 . . . . . 6 (𝐹:𝐴1-1-onto𝑥 ↔ (𝐹:𝐴onto𝑥 ∧ Fun 𝐹))
65anbi1i 624 . . . . 5 ((𝐹:𝐴1-1-onto𝑥𝑥𝐵) ↔ ((𝐹:𝐴onto𝑥 ∧ Fun 𝐹) ∧ 𝑥𝐵))
7 an32 646 . . . . 5 (((𝐹:𝐴onto𝑥 ∧ Fun 𝐹) ∧ 𝑥𝐵) ↔ ((𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
86, 7bitri 275 . . . 4 ((𝐹:𝐴1-1-onto𝑥𝑥𝐵) ↔ ((𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
98exbii 1845 . . 3 (∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵) ↔ ∃𝑥((𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
10 19.41v 1947 . . 3 (∃𝑥((𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹) ↔ (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
119, 10bitri 275 . 2 (∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵) ↔ (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
123, 4, 113bitr4i 303 1 (𝐹:𝐴1-1𝐵 ↔ ∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1776  wcel 2106  Vcvv 3478  wss 3963  ccnv 5688  Fun wfun 6557  wf 6559  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570
This theorem is referenced by:  domen  9001  uspgrsprfo  47992
  Copyright terms: Public domain W3C validator