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Theorem ffoss 5314
 Description: Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by set.mm contributors, 10-May-1998.)
Hypothesis
Ref Expression
f11o.1 F V
Assertion
Ref Expression
ffoss (F:A–→Bx(F:Aontox x B))
Distinct variable groups:   x,F   x,A   x,B

Proof of Theorem ffoss
StepHypRef Expression
1 df-f 4791 . . . 4 (F:A–→B ↔ (F Fn A ran F B))
2 dffn4 5275 . . . . 5 (F Fn AF:Aonto→ran F)
32anbi1i 676 . . . 4 ((F Fn A ran F B) ↔ (F:Aonto→ran F ran F B))
41, 3bitri 240 . . 3 (F:A–→B ↔ (F:Aonto→ran F ran F B))
5 f11o.1 . . . . 5 F V
65rnex 5107 . . . 4 ran F V
7 foeq3 5267 . . . . 5 (x = ran F → (F:AontoxF:Aonto→ran F))
8 sseq1 3292 . . . . 5 (x = ran F → (x B ↔ ran F B))
97, 8anbi12d 691 . . . 4 (x = ran F → ((F:Aontox x B) ↔ (F:Aonto→ran F ran F B)))
106, 9spcev 2946 . . 3 ((F:Aonto→ran F ran F B) → x(F:Aontox x B))
114, 10sylbi 187 . 2 (F:A–→Bx(F:Aontox x B))
12 fof 5269 . . . 4 (F:AontoxF:A–→x)
13 fss 5230 . . . 4 ((F:A–→x x B) → F:A–→B)
1412, 13sylan 457 . . 3 ((F:Aontox x B) → F:A–→B)
1514exlimiv 1634 . 2 (x(F:Aontox x B) → F:A–→B)
1611, 15impbii 180 1 (F:A–→Bx(F:Aontox x B))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  ran crn 4773   Fn wfn 4776  –→wf 4777  –onto→wfo 4779 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-ima 4727  df-rn 4786  df-f 4791  df-fo 4793 This theorem is referenced by:  f11o  5315
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