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Theorem fpwwe2lem10 10680
Description: Lemma for fpwwe2 10683. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.)
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 (𝜑𝐴𝑉)
fpwwe2.3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2.4 𝑋 = dom 𝑊
Assertion
Ref Expression
fpwwe2lem10 (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)   𝑉(𝑥,𝑦,𝑢,𝑟)

Proof of Theorem fpwwe2lem10
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))}
21relopabiv 5830 . . . . 5 Rel 𝑊
32a1i 11 . . . 4 (𝜑 → Rel 𝑊)
4 simprr 773 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))
5 fpwwe2.2 . . . . . . . . . . . . . . 15 (𝜑𝐴𝑉)
61, 5fpwwe2lem2 10672 . . . . . . . . . . . . . 14 (𝜑 → (𝑤𝑊𝑡 ↔ ((𝑤𝐴𝑡 ⊆ (𝑤 × 𝑤)) ∧ (𝑡 We 𝑤 ∧ ∀𝑦𝑤 [(𝑡 “ {𝑦}) / 𝑢](𝑢𝐹(𝑡 ∩ (𝑢 × 𝑢))) = 𝑦))))
76simprbda 498 . . . . . . . . . . . . 13 ((𝜑𝑤𝑊𝑡) → (𝑤𝐴𝑡 ⊆ (𝑤 × 𝑤)))
87simprd 495 . . . . . . . . . . . 12 ((𝜑𝑤𝑊𝑡) → 𝑡 ⊆ (𝑤 × 𝑤))
98adantrl 716 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑡 ⊆ (𝑤 × 𝑤))
109adantr 480 . . . . . . . . . 10 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑡 ⊆ (𝑤 × 𝑤))
11 dfss2 3969 . . . . . . . . . 10 (𝑡 ⊆ (𝑤 × 𝑤) ↔ (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
1210, 11sylib 218 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
134, 12eqtrd 2777 . . . . . . . 8 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑠 = 𝑡)
14 simprr 773 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))
151, 5fpwwe2lem2 10672 . . . . . . . . . . . . . 14 (𝜑 → (𝑤𝑊𝑠 ↔ ((𝑤𝐴𝑠 ⊆ (𝑤 × 𝑤)) ∧ (𝑠 We 𝑤 ∧ ∀𝑦𝑤 [(𝑠 “ {𝑦}) / 𝑢](𝑢𝐹(𝑠 ∩ (𝑢 × 𝑢))) = 𝑦))))
1615simprbda 498 . . . . . . . . . . . . 13 ((𝜑𝑤𝑊𝑠) → (𝑤𝐴𝑠 ⊆ (𝑤 × 𝑤)))
1716simprd 495 . . . . . . . . . . . 12 ((𝜑𝑤𝑊𝑠) → 𝑠 ⊆ (𝑤 × 𝑤))
1817adantrr 717 . . . . . . . . . . 11 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑠 ⊆ (𝑤 × 𝑤))
1918adantr 480 . . . . . . . . . 10 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑠 ⊆ (𝑤 × 𝑤))
20 dfss2 3969 . . . . . . . . . 10 (𝑠 ⊆ (𝑤 × 𝑤) ↔ (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
2119, 20sylib 218 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
2214, 21eqtr2d 2778 . . . . . . . 8 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑠 = 𝑡)
235adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝐴𝑉)
24 fpwwe2.3 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
2524adantlr 715 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
26 simprl 771 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑤𝑊𝑠)
27 simprr 773 . . . . . . . . 9 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑤𝑊𝑡)
281, 23, 25, 26, 27fpwwe2lem9 10679 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → ((𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤))) ∨ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))))
2913, 22, 28mpjaodan 961 . . . . . . 7 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑠 = 𝑡)
3029ex 412 . . . . . 6 (𝜑 → ((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
3130alrimiv 1927 . . . . 5 (𝜑 → ∀𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
3231alrimivv 1928 . . . 4 (𝜑 → ∀𝑤𝑠𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
33 dffun2 6571 . . . 4 (Fun 𝑊 ↔ (Rel 𝑊 ∧ ∀𝑤𝑠𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡)))
343, 32, 33sylanbrc 583 . . 3 (𝜑 → Fun 𝑊)
3534funfnd 6597 . 2 (𝜑𝑊 Fn dom 𝑊)
36 vex 3484 . . . . 5 𝑠 ∈ V
3736elrn 5904 . . . 4 (𝑠 ∈ ran 𝑊 ↔ ∃𝑤 𝑤𝑊𝑠)
382releldmi 5959 . . . . . . . . . . . 12 (𝑤𝑊𝑠𝑤 ∈ dom 𝑊)
3938adantl 481 . . . . . . . . . . 11 ((𝜑𝑤𝑊𝑠) → 𝑤 ∈ dom 𝑊)
40 elssuni 4937 . . . . . . . . . . 11 (𝑤 ∈ dom 𝑊𝑤 dom 𝑊)
4139, 40syl 17 . . . . . . . . . 10 ((𝜑𝑤𝑊𝑠) → 𝑤 dom 𝑊)
42 fpwwe2.4 . . . . . . . . . 10 𝑋 = dom 𝑊
4341, 42sseqtrrdi 4025 . . . . . . . . 9 ((𝜑𝑤𝑊𝑠) → 𝑤𝑋)
44 xpss12 5700 . . . . . . . . 9 ((𝑤𝑋𝑤𝑋) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
4543, 43, 44syl2anc 584 . . . . . . . 8 ((𝜑𝑤𝑊𝑠) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
4617, 45sstrd 3994 . . . . . . 7 ((𝜑𝑤𝑊𝑠) → 𝑠 ⊆ (𝑋 × 𝑋))
4746ex 412 . . . . . 6 (𝜑 → (𝑤𝑊𝑠𝑠 ⊆ (𝑋 × 𝑋)))
48 velpw 4605 . . . . . 6 (𝑠 ∈ 𝒫 (𝑋 × 𝑋) ↔ 𝑠 ⊆ (𝑋 × 𝑋))
4947, 48imbitrrdi 252 . . . . 5 (𝜑 → (𝑤𝑊𝑠𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5049exlimdv 1933 . . . 4 (𝜑 → (∃𝑤 𝑤𝑊𝑠𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5137, 50biimtrid 242 . . 3 (𝜑 → (𝑠 ∈ ran 𝑊𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5251ssrdv 3989 . 2 (𝜑 → ran 𝑊 ⊆ 𝒫 (𝑋 × 𝑋))
53 df-f 6565 . 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 395  w3a 1087  wal 1538   = wceq 1540  wex 1779  wcel 2108  wral 3061  [wsbc 3788  cin 3950  wss 3951  𝒫 cpw 4600  {csn 4626   cuni 4907   class class class wbr 5143  {copab 5205   We wwe 5636   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688  Rel wrel 5690  Fun wfun 6555   Fn wfn 6556  wf 6557  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-oi 9550
This theorem is referenced by:  fpwwe2lem12  10682  fpwwe2  10683
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