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Theorem fpwwe2lem11 9422
Description: Lemma for fpwwe2 9425. (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 5215 . . . . 5 Rel 𝑊
32a1i 11 . . . 4 (𝜑 → Rel 𝑊)
4 simprr 795 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))
5 fpwwe2.2 . . . . . . . . . . . . . . 15 (𝜑𝐴 ∈ V)
61, 5fpwwe2lem2 9414 . . . . . . . . . . . . . 14 (𝜑 → (𝑤𝑊𝑡 ↔ ((𝑤𝐴𝑡 ⊆ (𝑤 × 𝑤)) ∧ (𝑡 We 𝑤 ∧ ∀𝑦𝑤 [(𝑡 “ {𝑦}) / 𝑢](𝑢𝐹(𝑡 ∩ (𝑢 × 𝑢))) = 𝑦))))
76simprbda 652 . . . . . . . . . . . . 13 ((𝜑𝑤𝑊𝑡) → (𝑤𝐴𝑡 ⊆ (𝑤 × 𝑤)))
87simprd 479 . . . . . . . . . . . 12 ((𝜑𝑤𝑊𝑡) → 𝑡 ⊆ (𝑤 × 𝑤))
98adantrl 751 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑡 ⊆ (𝑤 × 𝑤))
109adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑡 ⊆ (𝑤 × 𝑤))
11 df-ss 3574 . . . . . . . . . 10 (𝑡 ⊆ (𝑤 × 𝑤) ↔ (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
1210, 11sylib 208 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
134, 12eqtrd 2655 . . . . . . . 8 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑠 = 𝑡)
14 simprr 795 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))
151, 5fpwwe2lem2 9414 . . . . . . . . . . . . . 14 (𝜑 → (𝑤𝑊𝑠 ↔ ((𝑤𝐴𝑠 ⊆ (𝑤 × 𝑤)) ∧ (𝑠 We 𝑤 ∧ ∀𝑦𝑤 [(𝑠 “ {𝑦}) / 𝑢](𝑢𝐹(𝑠 ∩ (𝑢 × 𝑢))) = 𝑦))))
1615simprbda 652 . . . . . . . . . . . . 13 ((𝜑𝑤𝑊𝑠) → (𝑤𝐴𝑠 ⊆ (𝑤 × 𝑤)))
1716simprd 479 . . . . . . . . . . . 12 ((𝜑𝑤𝑊𝑠) → 𝑠 ⊆ (𝑤 × 𝑤))
1817adantrr 752 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑠 ⊆ (𝑤 × 𝑤))
1918adantr 481 . . . . . . . . . 10 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑠 ⊆ (𝑤 × 𝑤))
20 df-ss 3574 . . . . . . . . . 10 (𝑠 ⊆ (𝑤 × 𝑤) ↔ (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
2119, 20sylib 208 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
2214, 21eqtr2d 2656 . . . . . . . 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 750 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
26 simprl 793 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑤𝑊𝑠)
27 simprr 795 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑤𝑊𝑡)
281, 23, 25, 26, 27fpwwe2lem10 9421 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → ((𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤))) ∨ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))))
2913, 22, 28mpjaodan 826 . . . . . . 7 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑠 = 𝑡)
3029ex 450 . . . . . 6 (𝜑 → ((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
3130alrimiv 1852 . . . . 5 (𝜑 → ∀𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
3231alrimivv 1853 . . . 4 (𝜑 → ∀𝑤𝑠𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
33 dffun2 5867 . . . 4 (Fun 𝑊 ↔ (Rel 𝑊 ∧ ∀𝑤𝑠𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡)))
343, 32, 33sylanbrc 697 . . 3 (𝜑 → Fun 𝑊)
35 funfn 5887 . . 3 (Fun 𝑊𝑊 Fn dom 𝑊)
3634, 35sylib 208 . 2 (𝜑𝑊 Fn dom 𝑊)
37 vex 3193 . . . . 5 𝑠 ∈ V
3837elrn 5336 . . . 4 (𝑠 ∈ ran 𝑊 ↔ ∃𝑤 𝑤𝑊𝑠)
392releldmi 5332 . . . . . . . . . . . 12 (𝑤𝑊𝑠𝑤 ∈ dom 𝑊)
4039adantl 482 . . . . . . . . . . 11 ((𝜑𝑤𝑊𝑠) → 𝑤 ∈ dom 𝑊)
41 elssuni 4440 . . . . . . . . . . 11 (𝑤 ∈ dom 𝑊𝑤 dom 𝑊)
4240, 41syl 17 . . . . . . . . . 10 ((𝜑𝑤𝑊𝑠) → 𝑤 dom 𝑊)
43 fpwwe2.4 . . . . . . . . . 10 𝑋 = dom 𝑊
4442, 43syl6sseqr 3637 . . . . . . . . 9 ((𝜑𝑤𝑊𝑠) → 𝑤𝑋)
45 xpss12 5196 . . . . . . . . 9 ((𝑤𝑋𝑤𝑋) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
4644, 44, 45syl2anc 692 . . . . . . . 8 ((𝜑𝑤𝑊𝑠) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
4717, 46sstrd 3598 . . . . . . 7 ((𝜑𝑤𝑊𝑠) → 𝑠 ⊆ (𝑋 × 𝑋))
4847ex 450 . . . . . 6 (𝜑 → (𝑤𝑊𝑠𝑠 ⊆ (𝑋 × 𝑋)))
49 selpw 4143 . . . . . 6 (𝑠 ∈ 𝒫 (𝑋 × 𝑋) ↔ 𝑠 ⊆ (𝑋 × 𝑋))
5048, 49syl6ibr 242 . . . . 5 (𝜑 → (𝑤𝑊𝑠𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5150exlimdv 1858 . . . 4 (𝜑 → (∃𝑤 𝑤𝑊𝑠𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5238, 51syl5bi 232 . . 3 (𝜑 → (𝑠 ∈ ran 𝑊𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5352ssrdv 3594 . 2 (𝜑 → ran 𝑊 ⊆ 𝒫 (𝑋 × 𝑋))
54 df-f 5861 . 2 (𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋) ↔ (𝑊 Fn dom 𝑊 ∧ ran 𝑊 ⊆ 𝒫 (𝑋 × 𝑋)))
5536, 53, 54sylanbrc 697 1 (𝜑𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  wral 2908  Vcvv 3190  [wsbc 3422  cin 3559  wss 3560  𝒫 cpw 4136  {csn 4155   cuni 4409   class class class wbr 4623  {copab 4682   We wwe 5042   × cxp 5082  ccnv 5083  dom cdm 5084  ran crn 5085  cima 5087  Rel wrel 5089  Fun wfun 5851   Fn wfn 5852  wf 5853  (class class class)co 6615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-wrecs 7367  df-recs 7428  df-oi 8375
This theorem is referenced by:  fpwwe2lem13  9424  fpwwe2  9425
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