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Theorem dff3 6358
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 6047 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
2 ffun 6035 . . . . . . . 8 (𝐹:𝐴𝐵 → Fun 𝐹)
3 fdm 6038 . . . . . . . . . 10 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
43eleq2d 2685 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑥 ∈ dom 𝐹𝑥𝐴))
54biimpar 502 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝐹)
6 funfvop 6315 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
72, 5, 6syl2an2r 875 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
8 df-br 4645 . . . . . . 7 (𝑥𝐹(𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
97, 8sylibr 224 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐹(𝐹𝑥))
10 fvex 6188 . . . . . . 7 (𝐹𝑥) ∈ V
11 breq2 4648 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝑥𝐹𝑦𝑥𝐹(𝐹𝑥)))
1210, 11spcev 3295 . . . . . 6 (𝑥𝐹(𝐹𝑥) → ∃𝑦 𝑥𝐹𝑦)
139, 12syl 17 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → ∃𝑦 𝑥𝐹𝑦)
14 funmo 5892 . . . . . . 7 (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
152, 14syl 17 . . . . . 6 (𝐹:𝐴𝐵 → ∃*𝑦 𝑥𝐹𝑦)
1615adantr 481 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → ∃*𝑦 𝑥𝐹𝑦)
17 eu5 2494 . . . . 5 (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
1813, 16, 17sylanbrc 697 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → ∃!𝑦 𝑥𝐹𝑦)
1918ralrimiva 2963 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)
201, 19jca 554 . 2 (𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
21 xpss 5216 . . . . . . . 8 (𝐴 × 𝐵) ⊆ (V × V)
22 sstr 3603 . . . . . . . 8 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (V × V)) → 𝐹 ⊆ (V × V))
2321, 22mpan2 706 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → 𝐹 ⊆ (V × V))
24 df-rel 5111 . . . . . . 7 (Rel 𝐹𝐹 ⊆ (V × V))
2523, 24sylibr 224 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → Rel 𝐹)
2625adantr 481 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → Rel 𝐹)
27 df-ral 2914 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥(𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦))
28 eumo 2497 . . . . . . . . . . . 12 (∃!𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
2928imim2i 16 . . . . . . . . . . 11 ((𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → (𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
3029adantl 482 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
31 df-br 4645 . . . . . . . . . . . . . . . 16 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
32 ssel 3589 . . . . . . . . . . . . . . . 16 (𝐹 ⊆ (𝐴 × 𝐵) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
3331, 32syl5bi 232 . . . . . . . . . . . . . . 15 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦 → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
34 opelxp1 5140 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
3533, 34syl6 35 . . . . . . . . . . . . . 14 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦𝑥𝐴))
3635exlimdv 1859 . . . . . . . . . . . . 13 (𝐹 ⊆ (𝐴 × 𝐵) → (∃𝑦 𝑥𝐹𝑦𝑥𝐴))
3736con3d 148 . . . . . . . . . . . 12 (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥𝐴 → ¬ ∃𝑦 𝑥𝐹𝑦))
38 exmo 2493 . . . . . . . . . . . . 13 (∃𝑦 𝑥𝐹𝑦 ∨ ∃*𝑦 𝑥𝐹𝑦)
3938ori 390 . . . . . . . . . . . 12 (¬ ∃𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
4037, 39syl6 35 . . . . . . . . . . 11 (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
4140adantr 481 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (¬ 𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
4230, 41pm2.61d 170 . . . . . . . . 9 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → ∃*𝑦 𝑥𝐹𝑦)
4342ex 450 . . . . . . . 8 (𝐹 ⊆ (𝐴 × 𝐵) → ((𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∃*𝑦 𝑥𝐹𝑦))
4443alimdv 1843 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥(𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
4527, 44syl5bi 232 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 → ∀𝑥∃*𝑦 𝑥𝐹𝑦))
4645imp 445 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
47 dffun6 5891 . . . . 5 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
4826, 46, 47sylanbrc 697 . . . 4 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → Fun 𝐹)
49 dmss 5312 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
50 dmxpss 5553 . . . . . . 7 dom (𝐴 × 𝐵) ⊆ 𝐴
5149, 50syl6ss 3607 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹𝐴)
52 breq1 4647 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐹𝑦𝑧𝐹𝑦))
5352eubidv 2488 . . . . . . . . 9 (𝑥 = 𝑧 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑧𝐹𝑦))
5453rspccv 3301 . . . . . . . 8 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧𝐴 → ∃!𝑦 𝑧𝐹𝑦))
55 euex 2492 . . . . . . . . 9 (∃!𝑦 𝑧𝐹𝑦 → ∃𝑦 𝑧𝐹𝑦)
56 vex 3198 . . . . . . . . . 10 𝑧 ∈ V
5756eldm 5310 . . . . . . . . 9 (𝑧 ∈ dom 𝐹 ↔ ∃𝑦 𝑧𝐹𝑦)
5855, 57sylibr 224 . . . . . . . 8 (∃!𝑦 𝑧𝐹𝑦𝑧 ∈ dom 𝐹)
5954, 58syl6 35 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧𝐴𝑧 ∈ dom 𝐹))
6059ssrdv 3601 . . . . . 6 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦𝐴 ⊆ dom 𝐹)
6151, 60anim12i 589 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → (dom 𝐹𝐴𝐴 ⊆ dom 𝐹))
62 eqss 3610 . . . . 5 (dom 𝐹 = 𝐴 ↔ (dom 𝐹𝐴𝐴 ⊆ dom 𝐹))
6361, 62sylibr 224 . . . 4 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → dom 𝐹 = 𝐴)
64 df-fn 5879 . . . 4 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6548, 63, 64sylanbrc 697 . . 3 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹 Fn 𝐴)
66 rnss 5343 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
67 rnxpss 5554 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
6866, 67syl6ss 3607 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
6968adantr 481 . . 3 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → ran 𝐹𝐵)
70 df-f 5880 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
7165, 69, 70sylanbrc 697 . 2 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹:𝐴𝐵)
7220, 71impbii 199 1 (𝐹:𝐴𝐵 ↔ (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1479   = wceq 1481  wex 1702  wcel 1988  ∃!weu 2468  ∃*wmo 2469  wral 2909  Vcvv 3195  wss 3567  cop 4174   class class class wbr 4644   × cxp 5102  dom cdm 5104  ran crn 5105  Rel wrel 5109  Fun wfun 5870   Fn wfn 5871  wf 5872  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884
This theorem is referenced by:  dff4  6359  seqomlem2  7531
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