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Theorem dffo5 6642
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 6641 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
2 rexex 3183 . . . . 5 (∃𝑥𝐴 𝑥𝐹𝑦 → ∃𝑥 𝑥𝐹𝑦)
32ralimi 3134 . . . 4 (∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦 → ∀𝑦𝐵𝑥 𝑥𝐹𝑦)
43anim2i 610 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
5 ffn 6293 . . . . . . . . 9 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
6 fnbr 6241 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
76ex 403 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
85, 7syl 17 . . . . . . . 8 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦𝑥𝐴))
98ancrd 547 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦 → (𝑥𝐴𝑥𝐹𝑦)))
109eximdv 1960 . . . . . 6 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥(𝑥𝐴𝑥𝐹𝑦)))
11 df-rex 3096 . . . . . 6 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
1210, 11syl6ibr 244 . . . . 5 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑥𝐹𝑦))
1312ralimdv 3145 . . . 4 (𝐹:𝐴𝐵 → (∀𝑦𝐵𝑥 𝑥𝐹𝑦 → ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
1413imdistani 564 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
154, 14impbii 201 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
161, 15bitri 267 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wex 1823  wcel 2107  wral 3090  wrex 3091   class class class wbr 4888   Fn wfn 6132  wf 6133  ontowfo 6135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-fo 6143  df-fv 6145
This theorem is referenced by: (None)
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