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Theorem dff4im 5664
Description: Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
Assertion
Ref Expression
dff4im (𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem dff4im
StepHypRef Expression
1 dff3im 5663 . 2 (𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
2 df-br 4006 . . . . . . . 8 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
3 ssel 3151 . . . . . . . . 9 (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
4 opelxp2 4663 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦𝐵)
53, 4syl6 33 . . . . . . . 8 (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹𝑦𝐵))
62, 5biimtrid 152 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦𝑦𝐵))
76pm4.71rd 394 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 ↔ (𝑦𝐵𝑥𝐹𝑦)))
87eubidv 2034 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦𝐵𝑥𝐹𝑦)))
9 df-reu 2462 . . . . 5 (∃!𝑦𝐵 𝑥𝐹𝑦 ↔ ∃!𝑦(𝑦𝐵𝑥𝐹𝑦))
108, 9bitr4di 198 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦𝐵 𝑥𝐹𝑦))
1110ralbidv 2477 . . 3 (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))
1211pm5.32i 454 . 2 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))
131, 12sylib 122 1 (𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦𝐵 𝑥𝐹𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  ∃!weu 2026  wcel 2148  wral 2455  ∃!wreu 2457  wss 3131  cop 3597   class class class wbr 4005   × cxp 4626  wf 5214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226
This theorem is referenced by: (None)
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