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

Theorem ffoss 7928
Description: Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
Hypothesis
Ref Expression
f11o.1 𝐹 ∈ V
Assertion
Ref Expression
ffoss (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ffoss
StepHypRef Expression
1 df-f 6544 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 dffn4 6808 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
32anbi1i 624 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵))
41, 3bitri 274 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵))
5 f11o.1 . . . . 5 𝐹 ∈ V
65rnex 7899 . . . 4 ran 𝐹 ∈ V
7 foeq3 6800 . . . . 5 (𝑥 = ran 𝐹 → (𝐹:𝐴onto𝑥𝐹:𝐴onto→ran 𝐹))
8 sseq1 4006 . . . . 5 (𝑥 = ran 𝐹 → (𝑥𝐵 ↔ ran 𝐹𝐵))
97, 8anbi12d 631 . . . 4 (𝑥 = ran 𝐹 → ((𝐹:𝐴onto𝑥𝑥𝐵) ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵)))
106, 9spcev 3596 . . 3 ((𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵) → ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
114, 10sylbi 216 . 2 (𝐹:𝐴𝐵 → ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
12 fof 6802 . . . 4 (𝐹:𝐴onto𝑥𝐹:𝐴𝑥)
13 fss 6731 . . . 4 ((𝐹:𝐴𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1412, 13sylan 580 . . 3 ((𝐹:𝐴onto𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1514exlimiv 1933 . 2 (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1611, 15impbii 208 1 (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  Vcvv 3474  wss 3947  ran crn 5676   Fn wfn 6535  wf 6536  ontowfo 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-cnv 5683  df-dm 5685  df-rn 5686  df-f 6544  df-fo 6546
This theorem is referenced by:  f11o  7929
  Copyright terms: Public domain W3C validator