MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dff4 Structured version   Visualization version   GIF version

Theorem dff4 6859
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 6858 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
2 df-br 5058 . . . . . . . 8 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
3 ssel 3958 . . . . . . . . 9 (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
4 opelxp2 5590 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦𝐵)
53, 4syl6 35 . . . . . . . 8 (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
62, 5syl5bi 243 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦𝑦𝐵))
76pm4.71rd 563 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 ↔ (𝑦𝐵𝑥𝐹𝑦)))
87eubidv 2665 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦𝐵𝑥𝐹𝑦)))
9 df-reu 3142 . . . . 5 (∃!𝑦𝐵 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦𝐵𝑥𝐹𝑦))
108, 9syl6bbr 290 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦𝐵 𝑥𝐹𝑦))
1110ralbidv 3194 . . 3 (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))
1211pm5.32i 575 . 2 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))
131, 12bitri 276 1 (𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2105  ∃!weu 2646  wral 3135  ∃!wreu 3137  wss 3933  cop 4563   class class class wbr 5057   × cxp 5546  wf 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356
This theorem is referenced by:  exfo  6863
  Copyright terms: Public domain W3C validator