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Theorem ffoss 5269
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 5006 . . . 4 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
2 dffn4 5223 . . . . 5 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
32anbi1i 446 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵))
41, 3bitri 182 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵))
5 f11o.1 . . . . 5 𝐹 ∈ V
65rnex 4688 . . . 4 ran 𝐹 ∈ V
7 foeq3 5215 . . . . 5 (𝑥 = ran 𝐹 → (𝐹:𝐴onto𝑥𝐹:𝐴onto→ran 𝐹))
8 sseq1 3045 . . . . 5 (𝑥 = ran 𝐹 → (𝑥𝐵 ↔ ran 𝐹𝐵))
97, 8anbi12d 457 . . . 4 (𝑥 = ran 𝐹 → ((𝐹:𝐴onto𝑥𝑥𝐵) ↔ (𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵)))
106, 9spcev 2713 . . 3 ((𝐹:𝐴onto→ran 𝐹 ∧ ran 𝐹𝐵) → ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
114, 10sylbi 119 . 2 (𝐹:𝐴𝐵 → ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
12 fof 5217 . . . 4 (𝐹:𝐴onto𝑥𝐹:𝐴𝑥)
13 fss 5157 . . . 4 ((𝐹:𝐴𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1412, 13sylan 277 . . 3 ((𝐹:𝐴onto𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1514exlimiv 1534 . 2 (∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵) → 𝐹:𝐴𝐵)
1611, 15impbii 124 1 (𝐹:𝐴𝐵 ↔ ∃𝑥(𝐹:𝐴onto𝑥𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  wss 2997  ran crn 4429   Fn wfn 4997  wf 4998  ontowfo 5000
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-cnv 4436  df-dm 4438  df-rn 4439  df-f 5006  df-fo 5008
This theorem is referenced by:  f11o  5270
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