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Theorem dff3 6844
 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 6509 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
2 ffun 6491 . . . . . . . 8 (𝐹:𝐴𝐵 → Fun 𝐹)
3 fdm 6496 . . . . . . . . . 10 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
43eleq2d 2875 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑥 ∈ dom 𝐹𝑥𝐴))
54biimpar 481 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝐹)
6 funfvop 6798 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
72, 5, 6syl2an2r 684 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
8 df-br 5032 . . . . . . 7 (𝑥𝐹(𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
97, 8sylibr 237 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐹(𝐹𝑥))
10 fvex 6659 . . . . . . 7 (𝐹𝑥) ∈ V
11 breq2 5035 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝑥𝐹𝑦𝑥𝐹(𝐹𝑥)))
1210, 11spcev 3555 . . . . . 6 (𝑥𝐹(𝐹𝑥) → ∃𝑦 𝑥𝐹𝑦)
139, 12syl 17 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → ∃𝑦 𝑥𝐹𝑦)
14 funmo 6341 . . . . . . 7 (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
152, 14syl 17 . . . . . 6 (𝐹:𝐴𝐵 → ∃*𝑦 𝑥𝐹𝑦)
1615adantr 484 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → ∃*𝑦 𝑥𝐹𝑦)
17 df-eu 2629 . . . . 5 (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
1813, 16, 17sylanbrc 586 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → ∃!𝑦 𝑥𝐹𝑦)
1918ralrimiva 3149 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)
201, 19jca 515 . 2 (𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
21 xpss 5536 . . . . . . . 8 (𝐴 × 𝐵) ⊆ (V × V)
22 sstr 3923 . . . . . . . 8 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (V × V)) → 𝐹 ⊆ (V × V))
2321, 22mpan2 690 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → 𝐹 ⊆ (V × V))
24 df-rel 5527 . . . . . . 7 (Rel 𝐹𝐹 ⊆ (V × V))
2523, 24sylibr 237 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → Rel 𝐹)
2625adantr 484 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → Rel 𝐹)
27 df-ral 3111 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥(𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦))
28 eumo 2638 . . . . . . . . . . . 12 (∃!𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
2928imim2i 16 . . . . . . . . . . 11 ((𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → (𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
3029adantl 485 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
31 df-br 5032 . . . . . . . . . . . . . . . 16 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
32 ssel 3908 . . . . . . . . . . . . . . . 16 (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
3331, 32syl5bi 245 . . . . . . . . . . . . . . 15 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
34 opelxp1 5561 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
3533, 34syl6 35 . . . . . . . . . . . . . 14 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦𝑥𝐴))
3635exlimdv 1934 . . . . . . . . . . . . 13 (𝐹 ⊆ (𝐴 × 𝐵) → (∃𝑦 𝑥𝐹𝑦𝑥𝐴))
3736con3d 155 . . . . . . . . . . . 12 (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥𝐴 → ¬ ∃𝑦 𝑥𝐹𝑦))
38 nexmo 2599 . . . . . . . . . . . 12 (¬ ∃𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
3937, 38syl6 35 . . . . . . . . . . 11 (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
4039adantr 484 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (¬ 𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
4130, 40pm2.61d 182 . . . . . . . . 9 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → ∃*𝑦 𝑥𝐹𝑦)
4241ex 416 . . . . . . . 8 (𝐹 ⊆ (𝐴 × 𝐵) → ((𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∃*𝑦 𝑥𝐹𝑦))
4342alimdv 1917 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥(𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
4427, 43syl5bi 245 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
4544imp 410 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
46 dffun6 6340 . . . . 5 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
4726, 45, 46sylanbrc 586 . . . 4 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → Fun 𝐹)
48 dmss 5736 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
49 dmxpss 5996 . . . . . . 7 dom (𝐴 × 𝐵) ⊆ 𝐴
5048, 49sstrdi 3927 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹𝐴)
51 breq1 5034 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐹𝑦𝑧𝐹𝑦))
5251eubidv 2647 . . . . . . . . 9 (𝑥 = 𝑧 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑧𝐹𝑦))
5352rspccv 3568 . . . . . . . 8 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧𝐴 → ∃!𝑦 𝑧𝐹𝑦))
54 euex 2637 . . . . . . . . 9 (∃!𝑦 𝑧𝐹𝑦 → ∃𝑦 𝑧𝐹𝑦)
55 vex 3444 . . . . . . . . . 10 𝑧 ∈ V
5655eldm 5734 . . . . . . . . 9 (𝑧 ∈ dom 𝐹 ↔ ∃𝑦 𝑧𝐹𝑦)
5754, 56sylibr 237 . . . . . . . 8 (∃!𝑦 𝑧𝐹𝑦𝑧 ∈ dom 𝐹)
5853, 57syl6 35 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧𝐴𝑧 ∈ dom 𝐹))
5958ssrdv 3921 . . . . . 6 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦𝐴 ⊆ dom 𝐹)
6050, 59anim12i 615 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → (dom 𝐹𝐴𝐴 ⊆ dom 𝐹))
61 eqss 3930 . . . . 5 (dom 𝐹 = 𝐴 ↔ (dom 𝐹𝐴𝐴 ⊆ dom 𝐹))
6260, 61sylibr 237 . . . 4 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → dom 𝐹 = 𝐴)
63 df-fn 6328 . . . 4 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6447, 62, 63sylanbrc 586 . . 3 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹 Fn 𝐴)
65 rnss 5774 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
66 rnxpss 5997 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
6765, 66sstrdi 3927 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
6867adantr 484 . . 3 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → ran 𝐹𝐵)
69 df-f 6329 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
7064, 68, 69sylanbrc 586 . 2 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹:𝐴𝐵)
7120, 70impbii 212 1 (𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃*wmo 2596  ∃!weu 2628  ∀wral 3106  Vcvv 3441   ⊆ wss 3881  ⟨cop 4531   class class class wbr 5031   × cxp 5518  dom cdm 5520  ran crn 5521  Rel wrel 5525  Fun wfun 6319   Fn wfn 6320  ⟶wf 6321  ‘cfv 6325 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333 This theorem is referenced by:  dff4  6845  seqomlem2  8073
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