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Theorem exfo 6417
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 6415 . . . 4 (𝑓:𝐴onto𝐵 ↔ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
2 dff4 6413 . . . . . 6 (𝑓:𝐴𝐵 ↔ (𝑓 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦))
32simprbi 479 . . . . 5 (𝑓:𝐴𝐵 → ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦)
43anim1i 591 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
51, 4sylbi 207 . . 3 (𝑓:𝐴onto𝐵 → (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
65eximi 1802 . 2 (∃𝑓 𝑓:𝐴onto𝐵 → ∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
7 brinxp 5215 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → (𝑥𝑓𝑦𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
87reubidva 3155 . . . . . . . . . . 11 (𝑥𝐴 → (∃!𝑦𝐵 𝑥𝑓𝑦 ↔ ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
98biimpd 219 . . . . . . . . . 10 (𝑥𝐴 → (∃!𝑦𝐵 𝑥𝑓𝑦 → ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
109ralimia 2979 . . . . . . . . 9 (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 → ∀𝑥𝐴 ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦)
11 inss2 3867 . . . . . . . . 9 (𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
1210, 11jctil 559 . . . . . . . 8 (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 → ((𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
13 dff4 6413 . . . . . . . 8 ((𝑓 ∩ (𝐴 × 𝐵)):𝐴𝐵 ↔ ((𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
1412, 13sylibr 224 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 → (𝑓 ∩ (𝐴 × 𝐵)):𝐴𝐵)
15 rninxp 5608 . . . . . . . 8 (ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥)
1615biimpri 218 . . . . . . 7 (∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥 → ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵)
1714, 16anim12i 589 . . . . . 6 ((∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ((𝑓 ∩ (𝐴 × 𝐵)):𝐴𝐵 ∧ ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵))
18 dffo2 6157 . . . . . 6 ((𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵 ↔ ((𝑓 ∩ (𝐴 × 𝐵)):𝐴𝐵 ∧ ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵))
1917, 18sylibr 224 . . . . 5 ((∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → (𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵)
20 vex 3234 . . . . . . 7 𝑓 ∈ V
2120inex1 4832 . . . . . 6 (𝑓 ∩ (𝐴 × 𝐵)) ∈ V
22 foeq1 6149 . . . . . 6 (𝑔 = (𝑓 ∩ (𝐴 × 𝐵)) → (𝑔:𝐴onto𝐵 ↔ (𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵))
2321, 22spcev 3331 . . . . 5 ((𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵 → ∃𝑔 𝑔:𝐴onto𝐵)
2419, 23syl 17 . . . 4 ((∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ∃𝑔 𝑔:𝐴onto𝐵)
2524exlimiv 1898 . . 3 (∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ∃𝑔 𝑔:𝐴onto𝐵)
26 foeq1 6149 . . . 4 (𝑔 = 𝑓 → (𝑔:𝐴onto𝐵𝑓:𝐴onto𝐵))
2726cbvexv 2311 . . 3 (∃𝑔 𝑔:𝐴onto𝐵 ↔ ∃𝑓 𝑓:𝐴onto𝐵)
2825, 27sylib 208 . 2 (∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ∃𝑓 𝑓:𝐴onto𝐵)
296, 28impbii 199 1 (∃𝑓 𝑓:𝐴onto𝐵 ↔ ∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  ∃!wreu 2943  cin 3606  wss 3607   class class class wbr 4685   × cxp 5141  ran crn 5144  wf 5922  ontowfo 5924
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934
This theorem is referenced by: (None)
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