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Theorem ffoss 7932
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:𝐴⟢𝐡 ↔ βˆƒπ‘₯(𝐹:𝐴–ontoβ†’π‘₯ ∧ π‘₯ βŠ† 𝐡))
Distinct variable groups:   π‘₯,𝐹   π‘₯,𝐴   π‘₯,𝐡

Proof of Theorem ffoss
StepHypRef Expression
1 df-f 6548 . . . 4 (𝐹:𝐴⟢𝐡 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† 𝐡))
2 dffn4 6812 . . . . 5 (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–ontoβ†’ran 𝐹)
32anbi1i 625 . . . 4 ((𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† 𝐡) ↔ (𝐹:𝐴–ontoβ†’ran 𝐹 ∧ ran 𝐹 βŠ† 𝐡))
41, 3bitri 275 . . 3 (𝐹:𝐴⟢𝐡 ↔ (𝐹:𝐴–ontoβ†’ran 𝐹 ∧ ran 𝐹 βŠ† 𝐡))
5 f11o.1 . . . . 5 𝐹 ∈ V
65rnex 7903 . . . 4 ran 𝐹 ∈ V
7 foeq3 6804 . . . . 5 (π‘₯ = ran 𝐹 β†’ (𝐹:𝐴–ontoβ†’π‘₯ ↔ 𝐹:𝐴–ontoβ†’ran 𝐹))
8 sseq1 4008 . . . . 5 (π‘₯ = ran 𝐹 β†’ (π‘₯ βŠ† 𝐡 ↔ ran 𝐹 βŠ† 𝐡))
97, 8anbi12d 632 . . . 4 (π‘₯ = ran 𝐹 β†’ ((𝐹:𝐴–ontoβ†’π‘₯ ∧ π‘₯ βŠ† 𝐡) ↔ (𝐹:𝐴–ontoβ†’ran 𝐹 ∧ ran 𝐹 βŠ† 𝐡)))
106, 9spcev 3597 . . 3 ((𝐹:𝐴–ontoβ†’ran 𝐹 ∧ ran 𝐹 βŠ† 𝐡) β†’ βˆƒπ‘₯(𝐹:𝐴–ontoβ†’π‘₯ ∧ π‘₯ βŠ† 𝐡))
114, 10sylbi 216 . 2 (𝐹:𝐴⟢𝐡 β†’ βˆƒπ‘₯(𝐹:𝐴–ontoβ†’π‘₯ ∧ π‘₯ βŠ† 𝐡))
12 fof 6806 . . . 4 (𝐹:𝐴–ontoβ†’π‘₯ β†’ 𝐹:𝐴⟢π‘₯)
13 fss 6735 . . . 4 ((𝐹:𝐴⟢π‘₯ ∧ π‘₯ βŠ† 𝐡) β†’ 𝐹:𝐴⟢𝐡)
1412, 13sylan 581 . . 3 ((𝐹:𝐴–ontoβ†’π‘₯ ∧ π‘₯ βŠ† 𝐡) β†’ 𝐹:𝐴⟢𝐡)
1514exlimiv 1934 . 2 (βˆƒπ‘₯(𝐹:𝐴–ontoβ†’π‘₯ ∧ π‘₯ βŠ† 𝐡) β†’ 𝐹:𝐴⟢𝐡)
1611, 15impbii 208 1 (𝐹:𝐴⟢𝐡 ↔ βˆƒπ‘₯(𝐹:𝐴–ontoβ†’π‘₯ ∧ π‘₯ βŠ† 𝐡))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3475   βŠ† wss 3949  ran crn 5678   Fn wfn 6539  βŸΆwf 6540  β€“ontoβ†’wfo 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-cnv 5685  df-dm 5687  df-rn 5688  df-f 6548  df-fo 6550
This theorem is referenced by:  f11o  7933
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