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Theorem exfo 7107
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 7105 . . . 4 (𝑓:𝐴onto𝐵 ↔ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
2 dff4 7103 . . . . . 6 (𝑓:𝐴𝐵 ↔ (𝑓 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦))
32simprbi 498 . . . . 5 (𝑓:𝐴𝐵 → ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦)
43anim1i 616 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
51, 4sylbi 216 . . 3 (𝑓:𝐴onto𝐵 → (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
65eximi 1838 . 2 (∃𝑓 𝑓:𝐴onto𝐵 → ∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
7 brinxp 5755 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → (𝑥𝑓𝑦𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
87reubidva 3393 . . . . . . . . . . 11 (𝑥𝐴 → (∃!𝑦𝐵 𝑥𝑓𝑦 ↔ ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
98biimpd 228 . . . . . . . . . 10 (𝑥𝐴 → (∃!𝑦𝐵 𝑥𝑓𝑦 → ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
109ralimia 3081 . . . . . . . . 9 (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 → ∀𝑥𝐴 ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦)
11 inss2 4230 . . . . . . . . 9 (𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
1210, 11jctil 521 . . . . . . . 8 (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 → ((𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
13 dff4 7103 . . . . . . . 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 6179 . . . . . . . 8 (ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥)
1615biimpri 227 . . . . . . 7 (∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥 → ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵)
1714, 16anim12i 614 . . . . . 6 ((∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ((𝑓 ∩ (𝐴 × 𝐵)):𝐴𝐵 ∧ ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵))
18 dffo2 6810 . . . . . 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 3479 . . . . . . 7 𝑓 ∈ V
2120inex1 5318 . . . . . 6 (𝑓 ∩ (𝐴 × 𝐵)) ∈ V
22 foeq1 6802 . . . . . 6 (𝑔 = (𝑓 ∩ (𝐴 × 𝐵)) → (𝑔:𝐴onto𝐵 ↔ (𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵))
2321, 22spcev 3597 . . . . 5 ((𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵 → ∃𝑔 𝑔:𝐴onto𝐵)
2419, 23syl 17 . . . 4 ((∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ∃𝑔 𝑔:𝐴onto𝐵)
2524exlimiv 1934 . . 3 (∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ∃𝑔 𝑔:𝐴onto𝐵)
26 foeq1 6802 . . . 4 (𝑔 = 𝑓 → (𝑔:𝐴onto𝐵𝑓:𝐴onto𝐵))
2726cbvexvw 2041 . . 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 397   = wceq 1542  wex 1782  wcel 2107  wral 3062  wrex 3071  ∃!wreu 3375  cin 3948  wss 3949   class class class wbr 5149   × cxp 5675  ran crn 5678  wf 6540  ontowfo 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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  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-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552
This theorem is referenced by: (None)
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