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Theorem dffo5 5642
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 5641 . 2 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
2 rexex 2516 . . . . 5 (∃𝑥𝐴 𝑥𝐹𝑦 → ∃𝑥 𝑥𝐹𝑦)
32ralimi 2533 . . . 4 (∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦 → ∀𝑦𝐵𝑥 𝑥𝐹𝑦)
43anim2i 340 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
5 ffn 5345 . . . . . . . . 9 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
6 fnbr 5298 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝑥𝐹𝑦) → 𝑥𝐴)
76ex 114 . . . . . . . . 9 (𝐹 Fn 𝐴 → (𝑥𝐹𝑦𝑥𝐴))
85, 7syl 14 . . . . . . . 8 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦𝑥𝐴))
98ancrd 324 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑥𝐹𝑦 → (𝑥𝐴𝑥𝐹𝑦)))
109eximdv 1873 . . . . . 6 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥(𝑥𝐴𝑥𝐹𝑦)))
11 df-rex 2454 . . . . . 6 (∃𝑥𝐴 𝑥𝐹𝑦 ↔ ∃𝑥(𝑥𝐴𝑥𝐹𝑦))
1210, 11syl6ibr 161 . . . . 5 (𝐹:𝐴𝐵 → (∃𝑥 𝑥𝐹𝑦 → ∃𝑥𝐴 𝑥𝐹𝑦))
1312ralimdv 2538 . . . 4 (𝐹:𝐴𝐵 → (∀𝑦𝐵𝑥 𝑥𝐹𝑦 → ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
1413imdistani 443 . . 3 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦) → (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦))
154, 14impbii 125 . 2 ((𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥𝐴 𝑥𝐹𝑦) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
161, 15bitri 183 1 (𝐹:𝐴onto𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵𝑥 𝑥𝐹𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wex 1485  wcel 2141  wral 2448  wrex 2449   class class class wbr 3987   Fn wfn 5191  wf 5192  ontowfo 5194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fo 5202  df-fv 5204
This theorem is referenced by: (None)
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