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Theorem f11o 7971
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 7970 . . 3 (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
32anbi1i 624 . 2 ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ↔ (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
4 df-f1 6566 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
5 dff1o3 6854 . . . . . 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 1848 . . 3 (∃𝑥(𝐹:𝐴1-1-onto𝑥𝑥𝐵) ↔ ∃𝑥((𝐹:𝐴onto𝑥𝑥𝐵) ∧ Fun 𝐹))
10 19.41v 1949 . . 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 1779  wcel 2108  Vcvv 3480  wss 3951  ccnv 5684  Fun wfun 6555  wf 6557  1-1wf1 6558  ontowfo 6559  1-1-ontowf1o 6560
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568
This theorem is referenced by:  domen  9002  uspgrsprfo  48064
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