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Theorem exfo 7125
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 7123 . . . 4 (𝑓:𝐴onto𝐵 ↔ (𝑓:𝐴𝐵 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
2 dff4 7121 . . . . . 6 (𝑓:𝐴𝐵 ↔ (𝑓 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦))
32simprbi 496 . . . . 5 (𝑓:𝐴𝐵 → ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦)
43anim1i 615 . . . 4 ((𝑓:𝐴𝐵 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
51, 4sylbi 217 . . 3 (𝑓:𝐴onto𝐵 → (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
65eximi 1832 . 2 (∃𝑓 𝑓:𝐴onto𝐵 → ∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
7 brinxp 5767 . . . . . . . . . . . 12 ((𝑥𝐴𝑦𝐵) → (𝑥𝑓𝑦𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
87reubidva 3394 . . . . . . . . . . 11 (𝑥𝐴 → (∃!𝑦𝐵 𝑥𝑓𝑦 ↔ ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
98biimpd 229 . . . . . . . . . 10 (𝑥𝐴 → (∃!𝑦𝐵 𝑥𝑓𝑦 → ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
109ralimia 3078 . . . . . . . . 9 (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 → ∀𝑥𝐴 ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦)
11 inss2 4246 . . . . . . . . 9 (𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵)
1210, 11jctil 519 . . . . . . . 8 (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 → ((𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
13 dff4 7121 . . . . . . . 8 ((𝑓 ∩ (𝐴 × 𝐵)):𝐴𝐵 ↔ ((𝑓 ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥(𝑓 ∩ (𝐴 × 𝐵))𝑦))
1412, 13sylibr 234 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 → (𝑓 ∩ (𝐴 × 𝐵)):𝐴𝐵)
15 rninxp 6201 . . . . . . . 8 (ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥)
1615biimpri 228 . . . . . . 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 6825 . . . . . 6 ((𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵 ↔ ((𝑓 ∩ (𝐴 × 𝐵)):𝐴𝐵 ∧ ran (𝑓 ∩ (𝐴 × 𝐵)) = 𝐵))
1917, 18sylibr 234 . . . . 5 ((∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → (𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵)
20 vex 3482 . . . . . . 7 𝑓 ∈ V
2120inex1 5323 . . . . . 6 (𝑓 ∩ (𝐴 × 𝐵)) ∈ V
22 foeq1 6817 . . . . . 6 (𝑔 = (𝑓 ∩ (𝐴 × 𝐵)) → (𝑔:𝐴onto𝐵 ↔ (𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵))
2321, 22spcev 3606 . . . . 5 ((𝑓 ∩ (𝐴 × 𝐵)):𝐴onto𝐵 → ∃𝑔 𝑔:𝐴onto𝐵)
2419, 23syl 17 . . . 4 ((∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ∃𝑔 𝑔:𝐴onto𝐵)
2524exlimiv 1928 . . 3 (∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ∃𝑔 𝑔:𝐴onto𝐵)
26 foeq1 6817 . . . 4 (𝑔 = 𝑓 → (𝑔:𝐴onto𝐵𝑓:𝐴onto𝐵))
2726cbvexvw 2034 . . 3 (∃𝑔 𝑔:𝐴onto𝐵 ↔ ∃𝑓 𝑓:𝐴onto𝐵)
2825, 27sylib 218 . 2 (∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥) → ∃𝑓 𝑓:𝐴onto𝐵)
296, 28impbii 209 1 (∃𝑓 𝑓:𝐴onto𝐵 ↔ ∃𝑓(∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝑓𝑦 ∧ ∀𝑥𝐵𝑦𝐴 𝑦𝑓𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wral 3059  wrex 3068  ∃!wreu 3376  cin 3962  wss 3963   class class class wbr 5148   × cxp 5687  ran crn 5690  wf 6559  ontowfo 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571
This theorem is referenced by: (None)
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