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Theorem dff3 6535
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 6221 . . 3 (𝐹:𝐴𝐵𝐹 ⊆ (𝐴 × 𝐵))
2 ffun 6209 . . . . . . . 8 (𝐹:𝐴𝐵 → Fun 𝐹)
3 fdm 6212 . . . . . . . . . 10 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
43eleq2d 2825 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑥 ∈ dom 𝐹𝑥𝐴))
54biimpar 503 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥 ∈ dom 𝐹)
6 funfvop 6492 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
72, 5, 6syl2an2r 911 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
8 df-br 4805 . . . . . . 7 (𝑥𝐹(𝐹𝑥) ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
97, 8sylibr 224 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐹(𝐹𝑥))
10 fvex 6362 . . . . . . 7 (𝐹𝑥) ∈ V
11 breq2 4808 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝑥𝐹𝑦𝑥𝐹(𝐹𝑥)))
1210, 11spcev 3440 . . . . . 6 (𝑥𝐹(𝐹𝑥) → ∃𝑦 𝑥𝐹𝑦)
139, 12syl 17 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → ∃𝑦 𝑥𝐹𝑦)
14 funmo 6065 . . . . . . 7 (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
152, 14syl 17 . . . . . 6 (𝐹:𝐴𝐵 → ∃*𝑦 𝑥𝐹𝑦)
1615adantr 472 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → ∃*𝑦 𝑥𝐹𝑦)
17 eu5 2633 . . . . 5 (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
1813, 16, 17sylanbrc 701 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → ∃!𝑦 𝑥𝐹𝑦)
1918ralrimiva 3104 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦)
201, 19jca 555 . 2 (𝐹:𝐴𝐵 → (𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦))
21 xpss 5282 . . . . . . . 8 (𝐴 × 𝐵) ⊆ (V × V)
22 sstr 3752 . . . . . . . 8 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (V × V)) → 𝐹 ⊆ (V × V))
2321, 22mpan2 709 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → 𝐹 ⊆ (V × V))
24 df-rel 5273 . . . . . . 7 (Rel 𝐹𝐹 ⊆ (V × V))
2523, 24sylibr 224 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → Rel 𝐹)
2625adantr 472 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → Rel 𝐹)
27 df-ral 3055 . . . . . . 7 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥(𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦))
28 eumo 2636 . . . . . . . . . . . 12 (∃!𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
2928imim2i 16 . . . . . . . . . . 11 ((𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → (𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
3029adantl 473 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
31 df-br 4805 . . . . . . . . . . . . . . . 16 (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
32 ssel 3738 . . . . . . . . . . . . . . . 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 5307 . . . . . . . . . . . . . . 15 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥𝐴)
3533, 34syl6 35 . . . . . . . . . . . . . 14 (𝐹 ⊆ (𝐴 × 𝐵) → (𝑥𝐹𝑦𝑥𝐴))
3635exlimdv 2010 . . . . . . . . . . . . 13 (𝐹 ⊆ (𝐴 × 𝐵) → (∃𝑦 𝑥𝐹𝑦𝑥𝐴))
3736con3d 148 . . . . . . . . . . . 12 (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥𝐴 → ¬ ∃𝑦 𝑥𝐹𝑦))
38 exmo 2632 . . . . . . . . . . . . 13 (∃𝑦 𝑥𝐹𝑦 ∨ ∃*𝑦 𝑥𝐹𝑦)
3938ori 389 . . . . . . . . . . . 12 (¬ ∃𝑦 𝑥𝐹𝑦 → ∃*𝑦 𝑥𝐹𝑦)
4037, 39syl6 35 . . . . . . . . . . 11 (𝐹 ⊆ (𝐴 × 𝐵) → (¬ 𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
4140adantr 472 . . . . . . . . . 10 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → (¬ 𝑥𝐴 → ∃*𝑦 𝑥𝐹𝑦))
4230, 41pm2.61d 170 . . . . . . . . 9 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ (𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦)) → ∃*𝑦 𝑥𝐹𝑦)
4342ex 449 . . . . . . . 8 (𝐹 ⊆ (𝐴 × 𝐵) → ((𝑥𝐴 → ∃!𝑦 𝑥𝐹𝑦) → ∃*𝑦 𝑥𝐹𝑦))
4443alimdv 1994 . . . . . . 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 444 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
47 dffun6 6064 . . . . 5 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
4826, 46, 47sylanbrc 701 . . . 4 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → Fun 𝐹)
49 dmss 5478 . . . . . . 7 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹 ⊆ dom (𝐴 × 𝐵))
50 dmxpss 5723 . . . . . . 7 dom (𝐴 × 𝐵) ⊆ 𝐴
5149, 50syl6ss 3756 . . . . . 6 (𝐹 ⊆ (𝐴 × 𝐵) → dom 𝐹𝐴)
52 breq1 4807 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐹𝑦𝑧𝐹𝑦))
5352eubidv 2627 . . . . . . . . 9 (𝑥 = 𝑧 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑧𝐹𝑦))
5453rspccv 3446 . . . . . . . 8 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦 → (𝑧𝐴 → ∃!𝑦 𝑧𝐹𝑦))
55 euex 2631 . . . . . . . . 9 (∃!𝑦 𝑧𝐹𝑦 → ∃𝑦 𝑧𝐹𝑦)
56 vex 3343 . . . . . . . . . 10 𝑧 ∈ V
5756eldm 5476 . . . . . . . . 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 3750 . . . . . 6 (∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦𝐴 ⊆ dom 𝐹)
6151, 60anim12i 591 . . . . 5 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → (dom 𝐹𝐴𝐴 ⊆ dom 𝐹))
62 eqss 3759 . . . . 5 (dom 𝐹 = 𝐴 ↔ (dom 𝐹𝐴𝐴 ⊆ dom 𝐹))
6361, 62sylibr 224 . . . 4 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → dom 𝐹 = 𝐴)
64 df-fn 6052 . . . 4 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
6548, 63, 64sylanbrc 701 . . 3 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → 𝐹 Fn 𝐴)
66 rnss 5509 . . . . 5 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹 ⊆ ran (𝐴 × 𝐵))
67 rnxpss 5724 . . . . 5 ran (𝐴 × 𝐵) ⊆ 𝐵
6866, 67syl6ss 3756 . . . 4 (𝐹 ⊆ (𝐴 × 𝐵) → ran 𝐹𝐵)
6968adantr 472 . . 3 ((𝐹 ⊆ (𝐴 × 𝐵) ∧ ∀𝑥𝐴 ∃!𝑦 𝑥𝐹𝑦) → ran 𝐹𝐵)
70 df-f 6053 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
7165, 69, 70sylanbrc 701 . 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 383  wal 1630   = wceq 1632  wex 1853  wcel 2139  ∃!weu 2607  ∃*wmo 2608  wral 3050  Vcvv 3340  wss 3715  cop 4327   class class class wbr 4804   × cxp 5264  dom cdm 5266  ran crn 5267  Rel wrel 5271  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057
This theorem is referenced by:  dff4  6536  seqomlem2  7715
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