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Theorem dff3 6689
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 6363 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
2 ffun 6347 . . . . . . . 8 (𝐹:𝐴𝐵 → Fun 𝐹)
3 fdm 6352 . . . . . . . . . 10 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
43eleq2d 2851 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑥 ∈ dom 𝐹𝑥𝐴))
54biimpar 470 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝐹)
6 funfvop 6645 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
72, 5, 6syl2an2r 672 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
8 df-br 4930 . . . . . . 7 (𝑥𝐹(𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
97, 8sylibr 226 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐹(𝐹𝑥))
10 fvex 6512 . . . . . . 7 (𝐹𝑥) ∈ V
11 breq2 4933 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝑥𝐹𝑦𝑥𝐹(𝐹𝑥)))
1210, 11spcev 3525 . . . . . 6 (𝑥𝐹(𝐹𝑥) → ∃𝑦 𝑥𝐹𝑦)
139, 12syl 17 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → ∃𝑦 𝑥𝐹𝑦)
14 funmo 6204 . . . . . . 7 (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
152, 14syl 17 . . . . . 6 (𝐹:𝐴𝐵 → ∃*𝑦 𝑥𝐹𝑦)
1615adantr 473 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → ∃*𝑦 𝑥𝐹𝑦)
17 df-eu 2584 . . . . 5 (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
1813, 16, 17sylanbrc 575 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → ∃!𝑦 𝑥𝐹𝑦)
1918ralrimiva 3132 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)
201, 19jca 504 . 2 (𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
21 xpss 5423 . . . . . . . 8 (𝐴 × 𝐵) ⊆ (V × V)
22 sstr 3866 . . . . . . . 8 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (V × V)) → 𝐹 ⊆ (V × V))
2321, 22mpan2 678 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → 𝐹 ⊆ (V × V))
24 df-rel 5414 . . . . . . 7 (Rel 𝐹𝐹 ⊆ (V × V))
2523, 24sylibr 226 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → Rel 𝐹)
2625adantr 473 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → Rel 𝐹)
27 df-ral 3093 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥(𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦))
28 eumo 2597 . . . . . . . . . . . 12 (∃!𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
2928imim2i 16 . . . . . . . . . . 11 ((𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → (𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
3029adantl 474 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
31 df-br 4930 . . . . . . . . . . . . . . . 16 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
32 ssel 3852 . . . . . . . . . . . . . . . 16 (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
3331, 32syl5bi 234 . . . . . . . . . . . . . . 15 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
34 opelxp1 5448 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
3533, 34syl6 35 . . . . . . . . . . . . . 14 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦𝑥𝐴))
3635exlimdv 1892 . . . . . . . . . . . . 13 (𝐹 ⊆ (𝐴 × 𝐵) → (∃𝑦 𝑥𝐹𝑦𝑥𝐴))
3736con3d 150 . . . . . . . . . . . 12 (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥𝐴 → ¬ ∃𝑦 𝑥𝐹𝑦))
38 nexmo 2548 . . . . . . . . . . . 12 (¬ ∃𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
3937, 38syl6 35 . . . . . . . . . . 11 (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
4039adantr 473 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (¬ 𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
4130, 40pm2.61d 172 . . . . . . . . 9 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → ∃*𝑦 𝑥𝐹𝑦)
4241ex 405 . . . . . . . 8 (𝐹 ⊆ (𝐴 × 𝐵) → ((𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∃*𝑦 𝑥𝐹𝑦))
4342alimdv 1875 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥(𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
4427, 43syl5bi 234 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
4544imp 398 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
46 dffun6 6203 . . . . 5 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
4726, 45, 46sylanbrc 575 . . . 4 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → Fun 𝐹)
48 dmss 5621 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
49 dmxpss 5868 . . . . . . 7 dom (𝐴 × 𝐵) ⊆ 𝐴
5048, 49syl6ss 3870 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹𝐴)
51 breq1 4932 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐹𝑦𝑧𝐹𝑦))
5251eubidv 2605 . . . . . . . . 9 (𝑥 = 𝑧 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑧𝐹𝑦))
5352rspccv 3532 . . . . . . . 8 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧𝐴 → ∃!𝑦 𝑧𝐹𝑦))
54 euex 2596 . . . . . . . . 9 (∃!𝑦 𝑧𝐹𝑦 → ∃𝑦 𝑧𝐹𝑦)
55 vex 3418 . . . . . . . . . 10 𝑧 ∈ V
5655eldm 5619 . . . . . . . . 9 (𝑧 ∈ dom 𝐹 ↔ ∃𝑦 𝑧𝐹𝑦)
5754, 56sylibr 226 . . . . . . . 8 (∃!𝑦 𝑧𝐹𝑦𝑧 ∈ dom 𝐹)
5853, 57syl6 35 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧𝐴𝑧 ∈ dom 𝐹))
5958ssrdv 3864 . . . . . 6 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦𝐴 ⊆ dom 𝐹)
6050, 59anim12i 603 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → (dom 𝐹𝐴𝐴 ⊆ dom 𝐹))
61 eqss 3873 . . . . 5 (dom 𝐹 = 𝐴 ↔ (dom 𝐹𝐴𝐴 ⊆ dom 𝐹))
6260, 61sylibr 226 . . . 4 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → dom 𝐹 = 𝐴)
63 df-fn 6191 . . . 4 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6447, 62, 63sylanbrc 575 . . 3 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹 Fn 𝐴)
65 rnss 5652 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
66 rnxpss 5869 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
6765, 66syl6ss 3870 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
6867adantr 473 . . 3 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → ran 𝐹𝐵)
69 df-f 6192 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
7064, 68, 69sylanbrc 575 . 2 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹:𝐴𝐵)
7120, 70impbii 201 1 (𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wal 1505   = wceq 1507  wex 1742  wcel 2050  ∃*wmo 2545  ∃!weu 2583  wral 3088  Vcvv 3415  wss 3829  cop 4447   class class class wbr 4929   × cxp 5405  dom cdm 5407  ran crn 5408  Rel wrel 5412  Fun wfun 6182   Fn wfn 6183  wf 6184  cfv 6188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196
This theorem is referenced by:  dff4  6690  seqomlem2  7890
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