MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fpwwe2lem11 Structured version   Visualization version   GIF version

Theorem fpwwe2lem11 10050
Description: Lemma for fpwwe2 10053. (Contributed by Mario Carneiro, 15-May-2015.)
Hypotheses
Ref Expression
fpwwe2.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2 (𝜑𝐴 ∈ V)
fpwwe2.3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2.4 𝑋 = dom 𝑊
Assertion
Ref Expression
fpwwe2lem11 (𝜑𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
Distinct variable groups:   𝑦,𝑢,𝑟,𝑥,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝜑,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑥   𝑊,𝑟,𝑢,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑦,𝑢)

Proof of Theorem fpwwe2lem11
Dummy variables 𝑠 𝑡 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fpwwe2.1 . . . . . 6 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
21relopabi 5687 . . . . 5 Rel 𝑊
32a1i 11 . . . 4 (𝜑 → Rel 𝑊)
4 simprr 769 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))
5 fpwwe2.2 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ V)
61, 5fpwwe2lem2 10042 . . . . . . . . . . . . . 14 (𝜑 → (𝑤𝑊𝑡 ↔ ((𝑤𝐴𝑡 ⊆ (𝑤 × 𝑤)) ∧ (𝑡 We 𝑤 ∧ ∀𝑦𝑤 [(𝑡 “ {𝑦}) / 𝑢](𝑢𝐹(𝑡 ∩ (𝑢 × 𝑢))) = 𝑦))))
76simprbda 499 . . . . . . . . . . . . 13 ((𝜑𝑤𝑊𝑡) → (𝑤𝐴𝑡 ⊆ (𝑤 × 𝑤)))
87simprd 496 . . . . . . . . . . . 12 ((𝜑𝑤𝑊𝑡) → 𝑡 ⊆ (𝑤 × 𝑤))
98adantrl 712 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑡 ⊆ (𝑤 × 𝑤))
109adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑡 ⊆ (𝑤 × 𝑤))
11 df-ss 3949 . . . . . . . . . 10 (𝑡 ⊆ (𝑤 × 𝑤) ↔ (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
1210, 11sylib 219 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
134, 12eqtrd 2853 . . . . . . . 8 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑠 = 𝑡)
14 simprr 769 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))
151, 5fpwwe2lem2 10042 . . . . . . . . . . . . . 14 (𝜑 → (𝑤𝑊𝑠 ↔ ((𝑤𝐴𝑠 ⊆ (𝑤 × 𝑤)) ∧ (𝑠 We 𝑤 ∧ ∀𝑦𝑤 [(𝑠 “ {𝑦}) / 𝑢](𝑢𝐹(𝑠 ∩ (𝑢 × 𝑢))) = 𝑦))))
1615simprbda 499 . . . . . . . . . . . . 13 ((𝜑𝑤𝑊𝑠) → (𝑤𝐴𝑠 ⊆ (𝑤 × 𝑤)))
1716simprd 496 . . . . . . . . . . . 12 ((𝜑𝑤𝑊𝑠) → 𝑠 ⊆ (𝑤 × 𝑤))
1817adantrr 713 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑠 ⊆ (𝑤 × 𝑤))
1918adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑠 ⊆ (𝑤 × 𝑤))
20 df-ss 3949 . . . . . . . . . 10 (𝑠 ⊆ (𝑤 × 𝑤) ↔ (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
2119, 20sylib 219 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
2214, 21eqtr2d 2854 . . . . . . . 8 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑠 = 𝑡)
235adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝐴 ∈ V)
24 fpwwe2.3 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
2524adantlr 711 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
26 simprl 767 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑤𝑊𝑠)
27 simprr 769 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑤𝑊𝑡)
281, 23, 25, 26, 27fpwwe2lem10 10049 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → ((𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤))) ∨ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))))
2913, 22, 28mpjaodan 952 . . . . . . 7 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑠 = 𝑡)
3029ex 413 . . . . . 6 (𝜑 → ((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
3130alrimiv 1919 . . . . 5 (𝜑 → ∀𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
3231alrimivv 1920 . . . 4 (𝜑 → ∀𝑤𝑠𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
33 dffun2 6358 . . . 4 (Fun 𝑊 ↔ (Rel 𝑊 ∧ ∀𝑤𝑠𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡)))
343, 32, 33sylanbrc 583 . . 3 (𝜑 → Fun 𝑊)
3534funfnd 6379 . 2 (𝜑𝑊 Fn dom 𝑊)
36 vex 3495 . . . . 5 𝑠 ∈ V
3736elrn 5815 . . . 4 (𝑠 ∈ ran 𝑊 ↔ ∃𝑤 𝑤𝑊𝑠)
382releldmi 5811 . . . . . . . . . . . 12 (𝑤𝑊𝑠𝑤 ∈ dom 𝑊)
3938adantl 482 . . . . . . . . . . 11 ((𝜑𝑤𝑊𝑠) → 𝑤 ∈ dom 𝑊)
40 elssuni 4859 . . . . . . . . . . 11 (𝑤 ∈ dom 𝑊𝑤 dom 𝑊)
4139, 40syl 17 . . . . . . . . . 10 ((𝜑𝑤𝑊𝑠) → 𝑤 dom 𝑊)
42 fpwwe2.4 . . . . . . . . . 10 𝑋 = dom 𝑊
4341, 42sseqtrrdi 4015 . . . . . . . . 9 ((𝜑𝑤𝑊𝑠) → 𝑤𝑋)
44 xpss12 5563 . . . . . . . . 9 ((𝑤𝑋𝑤𝑋) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
4543, 43, 44syl2anc 584 . . . . . . . 8 ((𝜑𝑤𝑊𝑠) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
4617, 45sstrd 3974 . . . . . . 7 ((𝜑𝑤𝑊𝑠) → 𝑠 ⊆ (𝑋 × 𝑋))
4746ex 413 . . . . . 6 (𝜑 → (𝑤𝑊𝑠𝑠 ⊆ (𝑋 × 𝑋)))
48 velpw 4543 . . . . . 6 (𝑠 ∈ 𝒫 (𝑋 × 𝑋) ↔ 𝑠 ⊆ (𝑋 × 𝑋))
4947, 48syl6ibr 253 . . . . 5 (𝜑 → (𝑤𝑊𝑠𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5049exlimdv 1925 . . . 4 (𝜑 → (∃𝑤 𝑤𝑊𝑠𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5137, 50syl5bi 243 . . 3 (𝜑 → (𝑠 ∈ ran 𝑊𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5251ssrdv 3970 . 2 (𝜑 → ran 𝑊 ⊆ 𝒫 (𝑋 × 𝑋))
53 df-f 6352 . 2 (𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋) ↔ (𝑊 Fn dom 𝑊 ∧ ran 𝑊 ⊆ 𝒫 (𝑋 × 𝑋)))
5435, 52, 53sylanbrc 583 1 (𝜑𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079  wal 1526   = wceq 1528  wex 1771  wcel 2105  wral 3135  Vcvv 3492  [wsbc 3769  cin 3932  wss 3933  𝒫 cpw 4535  {csn 4557   cuni 4830   class class class wbr 5057  {copab 5119   We wwe 5506   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551  Rel wrel 5553  Fun wfun 6342   Fn wfn 6343  wf 6344  (class class class)co 7145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-wrecs 7936  df-recs 7997  df-oi 8962
This theorem is referenced by:  fpwwe2lem13  10052  fpwwe2  10053
  Copyright terms: Public domain W3C validator