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Theorem fpwwe2lem10 10677
Description: Lemma for fpwwe2 10680. (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 5832 . . . . 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 10669 . . . . . . . . . . . . . 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 3980 . . . . . . . . . 10 (𝑡 ⊆ (𝑤 × 𝑤) ↔ (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
1210, 11sylib 218 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
134, 12eqtrd 2774 . . . . . . . 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 10669 . . . . . . . . . . . . . 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 3980 . . . . . . . . . 10 (𝑠 ⊆ (𝑤 × 𝑤) ↔ (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
2119, 20sylib 218 . . . . . . . . 9 (((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) ∧ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
2214, 21eqtr2d 2775 . . . . . . . 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 10676 . . . . . . . 8 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → ((𝑤𝑤𝑠 = (𝑡 ∩ (𝑤 × 𝑤))) ∨ (𝑤𝑤𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))))
2913, 22, 28mpjaodan 960 . . . . . . 7 ((𝜑 ∧ (𝑤𝑊𝑠𝑤𝑊𝑡)) → 𝑠 = 𝑡)
3029ex 412 . . . . . 6 (𝜑 → ((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
3130alrimiv 1924 . . . . 5 (𝜑 → ∀𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
3231alrimivv 1925 . . . 4 (𝜑 → ∀𝑤𝑠𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡))
33 dffun2 6572 . . . 4 (Fun 𝑊 ↔ (Rel 𝑊 ∧ ∀𝑤𝑠𝑡((𝑤𝑊𝑠𝑤𝑊𝑡) → 𝑠 = 𝑡)))
343, 32, 33sylanbrc 583 . . 3 (𝜑 → Fun 𝑊)
3534funfnd 6598 . 2 (𝜑𝑊 Fn dom 𝑊)
36 vex 3481 . . . . 5 𝑠 ∈ V
3736elrn 5906 . . . 4 (𝑠 ∈ ran 𝑊 ↔ ∃𝑤 𝑤𝑊𝑠)
382releldmi 5961 . . . . . . . . . . . 12 (𝑤𝑊𝑠𝑤 ∈ dom 𝑊)
3938adantl 481 . . . . . . . . . . 11 ((𝜑𝑤𝑊𝑠) → 𝑤 ∈ dom 𝑊)
40 elssuni 4941 . . . . . . . . . . 11 (𝑤 ∈ dom 𝑊𝑤 dom 𝑊)
4139, 40syl 17 . . . . . . . . . 10 ((𝜑𝑤𝑊𝑠) → 𝑤 dom 𝑊)
42 fpwwe2.4 . . . . . . . . . 10 𝑋 = dom 𝑊
4341, 42sseqtrrdi 4046 . . . . . . . . 9 ((𝜑𝑤𝑊𝑠) → 𝑤𝑋)
44 xpss12 5703 . . . . . . . . 9 ((𝑤𝑋𝑤𝑋) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
4543, 43, 44syl2anc 584 . . . . . . . 8 ((𝜑𝑤𝑊𝑠) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
4617, 45sstrd 4005 . . . . . . 7 ((𝜑𝑤𝑊𝑠) → 𝑠 ⊆ (𝑋 × 𝑋))
4746ex 412 . . . . . 6 (𝜑 → (𝑤𝑊𝑠𝑠 ⊆ (𝑋 × 𝑋)))
48 velpw 4609 . . . . . 6 (𝑠 ∈ 𝒫 (𝑋 × 𝑋) ↔ 𝑠 ⊆ (𝑋 × 𝑋))
4947, 48imbitrrdi 252 . . . . 5 (𝜑 → (𝑤𝑊𝑠𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5049exlimdv 1930 . . . 4 (𝜑 → (∃𝑤 𝑤𝑊𝑠𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5137, 50biimtrid 242 . . 3 (𝜑 → (𝑠 ∈ ran 𝑊𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
5251ssrdv 4000 . 2 (𝜑 → ran 𝑊 ⊆ 𝒫 (𝑋 × 𝑋))
53 df-f 6566 . 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 1086  wal 1534   = wceq 1536  wex 1775  wcel 2105  wral 3058  [wsbc 3790  cin 3961  wss 3962  𝒫 cpw 4604  {csn 4630   cuni 4911   class class class wbr 5147  {copab 5209   We wwe 5639   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689  cima 5691  Rel wrel 5693  Fun wfun 6556   Fn wfn 6557  wf 6558  (class class class)co 7430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-oi 9547
This theorem is referenced by:  fpwwe2lem12  10679  fpwwe2  10680
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