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Theorem exfo 6981
Description: A relation equivalent to the existence of an onto mapping. The right-hand 𝑓 is not necessarily a function. (Contributed by NM, 20-Mar-2007.)
Assertion
Ref Expression
exfo (∃𝑓 𝑓:𝐴onto𝐵 ↔ ∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝐵,𝑓,𝑥,𝑦

Proof of Theorem exfo
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 dffo4 6979 . . . 4 (𝑓:𝐴onto𝐵 ↔ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
2 dff4 6977 . . . . . 6 (𝑓:𝐴𝐵 ↔ (𝑓 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦))
32simprbi 497 . . . . 5 (𝑓:𝐴𝐵 → ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦)
43anim1i 615 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
51, 4sylbi 216 . . 3 (𝑓:𝐴onto𝐵 → (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
65eximi 1837 . 2 (∃𝑓 𝑓:𝐴onto𝐵 → ∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
7 brinxp 5665 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → (𝑥𝑓𝑦𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
87reubidva 3322 . . . . . . . . . . 11 (𝑥𝐴 → (∃!𝑦𝐵 𝑥𝑓𝑦 ↔ ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
98biimpd 228 . . . . . . . . . 10 (𝑥𝐴 → (∃!𝑦𝐵 𝑥𝑓𝑦 → ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
109ralimia 3085 . . . . . . . . 9 (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 → ∀𝑥𝐴 ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦)
11 inss2 4163 . . . . . . . . 9 (𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
1210, 11jctil 520 . . . . . . . 8 (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 → ((𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
13 dff4 6977 . . . . . . . 8 ((𝑓 ∩ (𝐴 × 𝐵)):𝐴𝐵 ↔ ((𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
1412, 13sylibr 233 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 → (𝑓 ∩ (𝐴 × 𝐵)):𝐴𝐵)
15 rninxp 6082 . . . . . . . 8 (ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥)
1615biimpri 227 . . . . . . 7 (∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥 → ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵)
1714, 16anim12i 613 . . . . . 6 ((∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ((𝑓 ∩ (𝐴 × 𝐵)):𝐴𝐵 ∧ ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵))
18 dffo2 6692 . . . . . 6 ((𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵 ↔ ((𝑓 ∩ (𝐴 × 𝐵)):𝐴𝐵 ∧ ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵))
1917, 18sylibr 233 . . . . 5 ((∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → (𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵)
20 vex 3436 . . . . . . 7 𝑓 ∈ V
2120inex1 5241 . . . . . 6 (𝑓 ∩ (𝐴 × 𝐵)) ∈ V
22 foeq1 6684 . . . . . 6 (𝑔 = (𝑓 ∩ (𝐴 × 𝐵)) → (𝑔:𝐴onto𝐵 ↔ (𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵))
2321, 22spcev 3545 . . . . 5 ((𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵 → ∃𝑔 𝑔:𝐴onto𝐵)
2419, 23syl 17 . . . 4 ((∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ∃𝑔 𝑔:𝐴onto𝐵)
2524exlimiv 1933 . . 3 (∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ∃𝑔 𝑔:𝐴onto𝐵)
26 foeq1 6684 . . . 4 (𝑔 = 𝑓 → (𝑔:𝐴onto𝐵𝑓:𝐴onto𝐵))
2726cbvexvw 2040 . . 3 (∃𝑔 𝑔:𝐴onto𝐵 ↔ ∃𝑓 𝑓:𝐴onto𝐵)
2825, 27sylib 217 . 2 (∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ∃𝑓 𝑓:𝐴onto𝐵)
296, 28impbii 208 1 (∃𝑓 𝑓:𝐴onto𝐵 ↔ ∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1539  wex 1782  wcel 2106  wral 3064  wrex 3065  ∃!wreu 3066  cin 3886  wss 3887   class class class wbr 5074   × cxp 5587  ran crn 5590  wf 6429  ontowfo 6431
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441
This theorem is referenced by: (None)
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