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Theorem ffoss 5101
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:ABx(𝐹:Aontox xB))
Distinct variable groups:   x,𝐹   x,A   x,B

Proof of Theorem ffoss
StepHypRef Expression
1 df-f 4849 . . . 4 (𝐹:AB ↔ (𝐹 Fn A ran 𝐹B))
2 dffn4 5055 . . . . 5 (𝐹 Fn A𝐹:Aonto→ran 𝐹)
32anbi1i 431 . . . 4 ((𝐹 Fn A ran 𝐹B) ↔ (𝐹:Aonto→ran 𝐹 ran 𝐹B))
41, 3bitri 173 . . 3 (𝐹:AB ↔ (𝐹:Aonto→ran 𝐹 ran 𝐹B))
5 f11o.1 . . . . 5 𝐹 V
65rnex 4542 . . . 4 ran 𝐹 V
7 foeq3 5047 . . . . 5 (x = ran 𝐹 → (𝐹:Aontox𝐹:Aonto→ran 𝐹))
8 sseq1 2960 . . . . 5 (x = ran 𝐹 → (xB ↔ ran 𝐹B))
97, 8anbi12d 442 . . . 4 (x = ran 𝐹 → ((𝐹:Aontox xB) ↔ (𝐹:Aonto→ran 𝐹 ran 𝐹B)))
106, 9spcev 2641 . . 3 ((𝐹:Aonto→ran 𝐹 ran 𝐹B) → x(𝐹:Aontox xB))
114, 10sylbi 114 . 2 (𝐹:ABx(𝐹:Aontox xB))
12 fof 5049 . . . 4 (𝐹:Aontox𝐹:Ax)
13 fss 4997 . . . 4 ((𝐹:Ax xB) → 𝐹:AB)
1412, 13sylan 267 . . 3 ((𝐹:Aontox xB) → 𝐹:AB)
1514exlimiv 1486 . 2 (x(𝐹:Aontox xB) → 𝐹:AB)
1611, 15impbii 117 1 (𝐹:ABx(𝐹:Aontox xB))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  wss 2911  ran crn 4289   Fn wfn 4840  wf 4841  ontowfo 4843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-cnv 4296  df-dm 4298  df-rn 4299  df-f 4849  df-fo 4851
This theorem is referenced by:  f11o  5102
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