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Theorem dff3 7119
Description: Alternate definition of a mapping. (Contributed by NM, 20-Mar-2007.)
Assertion
Ref Expression
dff3 (𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem dff3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 fssxp 6763 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
2 ffun 6739 . . . . . . . 8 (𝐹:𝐴𝐵 → Fun 𝐹)
3 fdm 6745 . . . . . . . . . 10 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
43eleq2d 2824 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑥 ∈ dom 𝐹𝑥𝐴))
54biimpar 477 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝐹)
6 funfvop 7069 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
72, 5, 6syl2an2r 685 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
8 df-br 5148 . . . . . . 7 (𝑥𝐹(𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
97, 8sylibr 234 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐹(𝐹𝑥))
10 fvex 6919 . . . . . . 7 (𝐹𝑥) ∈ V
11 breq2 5151 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝑥𝐹𝑦𝑥𝐹(𝐹𝑥)))
1210, 11spcev 3605 . . . . . 6 (𝑥𝐹(𝐹𝑥) → ∃𝑦 𝑥𝐹𝑦)
139, 12syl 17 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → ∃𝑦 𝑥𝐹𝑦)
14 funmo 6582 . . . . . . 7 (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
152, 14syl 17 . . . . . 6 (𝐹:𝐴𝐵 → ∃*𝑦 𝑥𝐹𝑦)
1615adantr 480 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → ∃*𝑦 𝑥𝐹𝑦)
17 df-eu 2566 . . . . 5 (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
1813, 16, 17sylanbrc 583 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → ∃!𝑦 𝑥𝐹𝑦)
1918ralrimiva 3143 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)
201, 19jca 511 . 2 (𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
21 xpss 5704 . . . . . . . 8 (𝐴 × 𝐵) ⊆ (V × V)
22 sstr 4003 . . . . . . . 8 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (V × V)) → 𝐹 ⊆ (V × V))
2321, 22mpan2 691 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → 𝐹 ⊆ (V × V))
24 df-rel 5695 . . . . . . 7 (Rel 𝐹𝐹 ⊆ (V × V))
2523, 24sylibr 234 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → Rel 𝐹)
2625adantr 480 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → Rel 𝐹)
27 df-ral 3059 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥(𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦))
28 eumo 2575 . . . . . . . . . . . 12 (∃!𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
2928imim2i 16 . . . . . . . . . . 11 ((𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → (𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
3029adantl 481 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
31 df-br 5148 . . . . . . . . . . . . . . . 16 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
32 ssel 3988 . . . . . . . . . . . . . . . 16 (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
3331, 32biimtrid 242 . . . . . . . . . . . . . . 15 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
34 opelxp1 5730 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
3533, 34syl6 35 . . . . . . . . . . . . . 14 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦𝑥𝐴))
3635exlimdv 1930 . . . . . . . . . . . . 13 (𝐹 ⊆ (𝐴 × 𝐵) → (∃𝑦 𝑥𝐹𝑦𝑥𝐴))
3736con3d 152 . . . . . . . . . . . 12 (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥𝐴 → ¬ ∃𝑦 𝑥𝐹𝑦))
38 nexmo 2538 . . . . . . . . . . . 12 (¬ ∃𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
3937, 38syl6 35 . . . . . . . . . . 11 (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
4039adantr 480 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (¬ 𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
4130, 40pm2.61d 179 . . . . . . . . 9 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → ∃*𝑦 𝑥𝐹𝑦)
4241ex 412 . . . . . . . 8 (𝐹 ⊆ (𝐴 × 𝐵) → ((𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∃*𝑦 𝑥𝐹𝑦))
4342alimdv 1913 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥(𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
4427, 43biimtrid 242 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
4544imp 406 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
46 dffun6 6575 . . . . 5 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
4726, 45, 46sylanbrc 583 . . . 4 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → Fun 𝐹)
48 dmss 5915 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
49 dmxpss 6192 . . . . . . 7 dom (𝐴 × 𝐵) ⊆ 𝐴
5048, 49sstrdi 4007 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹𝐴)
51 breq1 5150 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐹𝑦𝑧𝐹𝑦))
5251eubidv 2583 . . . . . . . . 9 (𝑥 = 𝑧 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑧𝐹𝑦))
5352rspccv 3618 . . . . . . . 8 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧𝐴 → ∃!𝑦 𝑧𝐹𝑦))
54 euex 2574 . . . . . . . . 9 (∃!𝑦 𝑧𝐹𝑦 → ∃𝑦 𝑧𝐹𝑦)
55 vex 3481 . . . . . . . . . 10 𝑧 ∈ V
5655eldm 5913 . . . . . . . . 9 (𝑧 ∈ dom 𝐹 ↔ ∃𝑦 𝑧𝐹𝑦)
5754, 56sylibr 234 . . . . . . . 8 (∃!𝑦 𝑧𝐹𝑦𝑧 ∈ dom 𝐹)
5853, 57syl6 35 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧𝐴𝑧 ∈ dom 𝐹))
5958ssrdv 4000 . . . . . 6 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦𝐴 ⊆ dom 𝐹)
6050, 59anim12i 613 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → (dom 𝐹𝐴𝐴 ⊆ dom 𝐹))
61 eqss 4010 . . . . 5 (dom 𝐹 = 𝐴 ↔ (dom 𝐹𝐴𝐴 ⊆ dom 𝐹))
6260, 61sylibr 234 . . . 4 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → dom 𝐹 = 𝐴)
63 df-fn 6565 . . . 4 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6447, 62, 63sylanbrc 583 . . 3 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹 Fn 𝐴)
65 rnss 5952 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
66 rnxpss 6193 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
6765, 66sstrdi 4007 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
6867adantr 480 . . 3 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → ran 𝐹𝐵)
69 df-f 6566 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
7064, 68, 69sylanbrc 583 . 2 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹:𝐴𝐵)
7120, 70impbii 209 1 (𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1534   = wceq 1536  wex 1775  wcel 2105  ∃*wmo 2535  ∃!weu 2565  wral 3058  Vcvv 3477  wss 3962  cop 4636   class class class wbr 5147   × cxp 5686  dom cdm 5688  ran crn 5689  Rel wrel 5693  Fun wfun 6556   Fn wfn 6557  wf 6558  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570
This theorem is referenced by:  dff4  7120  seqomlem2  8489
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