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Theorem dffo5 7108
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 7107 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
2 rexex 3072 . . . . 5 (∃𝑥𝐴 𝑥𝐹𝑦 → ∃𝑥 𝑥𝐹𝑦)
32ralimi 3079 . . . 4 (∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦 → ∀𝑦𝐵𝑥 𝑥𝐹𝑦)
43anim2i 616 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
5 ffn 6716 . . . . . . . . 9 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
6 fnbr 6656 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
76ex 412 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
85, 7syl 17 . . . . . . . 8 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦𝑥𝐴))
98ancrd 551 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦 → (𝑥𝐴𝑥𝐹𝑦)))
109eximdv 1913 . . . . . 6 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥(𝑥𝐴𝑥𝐹𝑦)))
11 df-rex 3067 . . . . . 6 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
1210, 11imbitrrdi 251 . . . . 5 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑥𝐹𝑦))
1312ralimdv 3165 . . . 4 (𝐹:𝐴𝐵 → (∀𝑦𝐵𝑥 𝑥𝐹𝑦 → ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
1413imdistani 568 . . 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 395  wex 1774  wcel 2099  wral 3057  wrex 3066   class class class wbr 5142   Fn wfn 6537  wf 6538  ontowfo 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550
This theorem is referenced by: (None)
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