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Theorem ffoss 7970
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 6565 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 dffn4 6826 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
32anbi1i 624 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵))
41, 3bitri 275 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵))
5 f11o.1 . . . . 5 𝐹 ∈ V
65rnex 7932 . . . 4 ran 𝐹 ∈ V
7 foeq3 6818 . . . . 5 (𝑥 = ran 𝐹 → (𝐹:𝐴onto𝑥𝐹:𝐴onto→ran 𝐹))
8 sseq1 4009 . . . . 5 (𝑥 = ran 𝐹 → (𝑥𝐵 ↔ ran 𝐹𝐵))
97, 8anbi12d 632 . . . 4 (𝑥 = ran 𝐹 → ((𝐹:𝐴onto𝑥𝑥𝐵) ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵)))
106, 9spcev 3606 . . 3 ((𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵) → ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
114, 10sylbi 217 . 2 (𝐹:𝐴𝐵 → ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
12 fof 6820 . . . 4 (𝐹:𝐴onto𝑥𝐹:𝐴𝑥)
13 fss 6752 . . . 4 ((𝐹:𝐴𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1412, 13sylan 580 . . 3 ((𝐹:𝐴onto𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1514exlimiv 1930 . 2 (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1611, 15impbii 209 1 (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  wss 3951  ran crn 5686   Fn wfn 6556  wf 6557  ontowfo 6559
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-fo 6567
This theorem is referenced by:  f11o  7971
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