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Theorem ffoss 5367
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 5097 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 dffn4 5321 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
32anbi1i 453 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵))
41, 3bitri 183 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵))
5 f11o.1 . . . . 5 𝐹 ∈ V
65rnex 4776 . . . 4 ran 𝐹 ∈ V
7 foeq3 5313 . . . . 5 (𝑥 = ran 𝐹 → (𝐹:𝐴onto𝑥𝐹:𝐴onto→ran 𝐹))
8 sseq1 3090 . . . . 5 (𝑥 = ran 𝐹 → (𝑥𝐵 ↔ ran 𝐹𝐵))
97, 8anbi12d 464 . . . 4 (𝑥 = ran 𝐹 → ((𝐹:𝐴onto𝑥𝑥𝐵) ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵)))
106, 9spcev 2754 . . 3 ((𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵) → ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
114, 10sylbi 120 . 2 (𝐹:𝐴𝐵 → ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
12 fof 5315 . . . 4 (𝐹:𝐴onto𝑥𝐹:𝐴𝑥)
13 fss 5254 . . . 4 ((𝐹:𝐴𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1412, 13sylan 281 . . 3 ((𝐹:𝐴onto𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1514exlimiv 1562 . 2 (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1611, 15impbii 125 1 (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1316  wex 1453  wcel 1465  Vcvv 2660  wss 3041  ran crn 4510   Fn wfn 5088  wf 5089  ontowfo 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-cnv 4517  df-dm 4519  df-rn 4520  df-f 5097  df-fo 5099
This theorem is referenced by:  f11o  5368
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