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Theorem dffo5 7058
Description: Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
Assertion
Ref Expression
dffo5 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem dffo5
StepHypRef Expression
1 dffo4 7057 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
2 rexex 3076 . . . . 5 (∃𝑥𝐴 𝑥𝐹𝑦 → ∃𝑥 𝑥𝐹𝑦)
32ralimi 3083 . . . 4 (∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦 → ∀𝑦𝐵𝑥 𝑥𝐹𝑦)
43anim2i 618 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
5 ffn 6672 . . . . . . . . 9 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
6 fnbr 6614 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
76ex 414 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
85, 7syl 17 . . . . . . . 8 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦𝑥𝐴))
98ancrd 553 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦 → (𝑥𝐴𝑥𝐹𝑦)))
109eximdv 1921 . . . . . 6 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥(𝑥𝐴𝑥𝐹𝑦)))
11 df-rex 3071 . . . . . 6 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
1210, 11syl6ibr 252 . . . . 5 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑥𝐹𝑦))
1312ralimdv 3163 . . . 4 (𝐹:𝐴𝐵 → (∀𝑦𝐵𝑥 𝑥𝐹𝑦 → ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
1413imdistani 570 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
154, 14impbii 208 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
161, 15bitri 275 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wex 1782  wcel 2107  wral 3061  wrex 3070   class class class wbr 5109   Fn wfn 6495  wf 6496  ontowfo 6498
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 5260  ax-nul 5267  ax-pr 5388
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508
This theorem is referenced by: (None)
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