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Theorem fpwwe2lem11 9447
Description: Lemma for fpwwe2 9450. (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 5234 . . . . 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 9439 . . . . . . . . . . . . . 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 3581 . . . . . . . . . 10 (𝑡 ⊆ (𝑤 × 𝑤) ↔ (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
1210, 11sylib 208 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
134, 12eqtrd 2654 . . . . . . . 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 9439 . . . . . . . . . . . . . 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 3581 . . . . . . . . . 10 (𝑠 ⊆ (𝑤 × 𝑤) ↔ (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
2119, 20sylib 208 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
2214, 21eqtr2d 2655 . . . . . . . 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 9446 . . . . . . . 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 1853 . . . . 5 (𝜑 → ∀𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
3231alrimivv 1854 . . . 4 (𝜑 → ∀𝑤𝑠𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
33 dffun2 5886 . . . 4 (Fun 𝑊 ↔ (Rel 𝑊 ∧ ∀𝑤𝑠𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡)))
343, 32, 33sylanbrc 697 . . 3 (𝜑 → Fun 𝑊)
35 funfn 5906 . . 3 (Fun 𝑊𝑊 Fn dom 𝑊)
3634, 35sylib 208 . 2 (𝜑𝑊 Fn dom 𝑊)
37 vex 3198 . . . . 5 𝑠 ∈ V
3837elrn 5355 . . . 4 (𝑠 ∈ ran 𝑊 ↔ ∃𝑤 𝑤𝑊𝑠)
392releldmi 5351 . . . . . . . . . . . 12 (𝑤𝑊𝑠𝑤 ∈ dom 𝑊)
4039adantl 482 . . . . . . . . . . 11 ((𝜑𝑤𝑊𝑠) → 𝑤 ∈ dom 𝑊)
41 elssuni 4458 . . . . . . . . . . 11 (𝑤 ∈ dom 𝑊𝑤 dom 𝑊)
4240, 41syl 17 . . . . . . . . . 10 ((𝜑𝑤𝑊𝑠) → 𝑤 dom 𝑊)
43 fpwwe2.4 . . . . . . . . . 10 𝑋 = dom 𝑊
4442, 43syl6sseqr 3644 . . . . . . . . 9 ((𝜑𝑤𝑊𝑠) → 𝑤𝑋)
45 xpss12 5215 . . . . . . . . 9 ((𝑤𝑋𝑤𝑋) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
4644, 44, 45syl2anc 692 . . . . . . . 8 ((𝜑𝑤𝑊𝑠) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
4717, 46sstrd 3605 . . . . . . 7 ((𝜑𝑤𝑊𝑠) → 𝑠 ⊆ (𝑋 × 𝑋))
4847ex 450 . . . . . 6 (𝜑 → (𝑤𝑊𝑠𝑠 ⊆ (𝑋 × 𝑋)))
49 selpw 4156 . . . . . 6 (𝑠 ∈ 𝒫 (𝑋 × 𝑋) ↔ 𝑠 ⊆ (𝑋 × 𝑋))
5048, 49syl6ibr 242 . . . . 5 (𝜑 → (𝑤𝑊𝑠𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5150exlimdv 1859 . . . 4 (𝜑 → (∃𝑤 𝑤𝑊𝑠𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5238, 51syl5bi 232 . . 3 (𝜑 → (𝑠 ∈ ran 𝑊𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5352ssrdv 3601 . 2 (𝜑 → ran 𝑊 ⊆ 𝒫 (𝑋 × 𝑋))
54 df-f 5880 . 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 1479   = wceq 1481  wex 1702  wcel 1988  wral 2909  Vcvv 3195  [wsbc 3429  cin 3566  wss 3567  𝒫 cpw 4149  {csn 4168   cuni 4427   class class class wbr 4644  {copab 4703   We wwe 5062   × cxp 5102  ccnv 5103  dom cdm 5104  ran crn 5105  cima 5107  Rel wrel 5109  Fun wfun 5870   Fn wfn 5871  wf 5872  (class class class)co 6635
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-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  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-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-wrecs 7392  df-recs 7453  df-oi 8400
This theorem is referenced by:  fpwwe2lem13  9449  fpwwe2  9450
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