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Theorem dff4 6865
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 6864 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
2 df-br 5064 . . . . . . . 8 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
3 ssel 3965 . . . . . . . . 9 (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
4 opelxp2 5596 . . . . . . . . 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 2670 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦𝐵𝑥𝐹𝑦)))
9 df-reu 3150 . . . . 5 (∃!𝑦𝐵 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦𝐵𝑥𝐹𝑦))
108, 9syl6bbr 290 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦𝐵 𝑥𝐹𝑦))
1110ralbidv 3202 . . 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 2107  ∃!weu 2651  wral 3143  ∃!wreu 3145  wss 3940  cop 4570   class class class wbr 5063   × cxp 5552  wf 6350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362
This theorem is referenced by:  exfo  6869
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