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Theorem dff4 7120
Description: Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
Assertion
Ref Expression
dff4 (𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem dff4
StepHypRef Expression
1 dff3 7119 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
2 df-br 5143 . . . . . . . 8 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
3 ssel 3976 . . . . . . . . 9 (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
4 opelxp2 5727 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦𝐵)
53, 4syl6 35 . . . . . . . 8 (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
62, 5biimtrid 242 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦𝑦𝐵))
76pm4.71rd 562 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 ↔ (𝑦𝐵𝑥𝐹𝑦)))
87eubidv 2585 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦𝐵𝑥𝐹𝑦)))
9 df-reu 3380 . . . . 5 (∃!𝑦𝐵 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦𝐵𝑥𝐹𝑦))
108, 9bitr4di 289 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦𝐵 𝑥𝐹𝑦))
1110ralbidv 3177 . . 3 (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))
1211pm5.32i 574 . 2 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))
131, 12bitri 275 1 (𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2107  ∃!weu 2567  wral 3060  ∃!wreu 3377  wss 3950  cop 4631   class class class wbr 5142   × cxp 5682  wf 6556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568
This theorem is referenced by:  exfo  7124
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