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Theorem canthnumlem 10688
Description: Lemma for canthnum 10689. (Contributed by Mario Carneiro, 19-May-2015.)
Hypotheses
Ref Expression
canth4.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
canth4.2 𝐵 = dom 𝑊
canth4.3 𝐶 = ((𝑊𝐵) “ {(𝐹𝐵)})
Assertion
Ref Expression
canthnumlem (𝐴𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
Distinct variable groups:   𝑥,𝑟,𝑦,𝐴   𝐵,𝑟,𝑥,𝑦   𝐹,𝑟,𝑥,𝑦   𝑉,𝑟,𝑥,𝑦   𝑦,𝐶   𝑊,𝑟,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑟)

Proof of Theorem canthnumlem
StepHypRef Expression
1 f1f 6804 . . . . 5 (𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2 ssid 4006 . . . . . 6 (𝒫 𝐴 ∩ dom card) ⊆ (𝒫 𝐴 ∩ dom card)
3 canth4.1 . . . . . . 7 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
4 canth4.2 . . . . . . 7 𝐵 = dom 𝑊
5 canth4.3 . . . . . . 7 𝐶 = ((𝑊𝐵) “ {(𝐹𝐵)})
63, 4, 5canth4 10687 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ (𝒫 𝐴 ∩ dom card)) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
72, 6mp3an3 1452 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
81, 7sylan2 593 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
98simp3d 1145 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐹𝐵) = (𝐹𝐶))
10 simpr 484 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
118simp1d 1143 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝐴)
12 elpw2g 5333 . . . . . . 7 (𝐴𝑉 → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1312adantr 480 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1411, 13mpbird 257 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ 𝒫 𝐴)
15 eqid 2737 . . . . . . . . . . . 12 𝐵 = 𝐵
16 eqid 2737 . . . . . . . . . . . 12 (𝑊𝐵) = (𝑊𝐵)
1715, 16pm3.2i 470 . . . . . . . . . . 11 (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))
18 simpl 482 . . . . . . . . . . . 12 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐴𝑉)
1910, 1syl 17 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2019ffvelcdmda 7104 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹𝑥) ∈ 𝐴)
213, 18, 20, 4fpwwe 10686 . . . . . . . . . . 11 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))))
2217, 21mpbiri 258 . . . . . . . . . 10 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵))
2322simpld 494 . . . . . . . . 9 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝑊(𝑊𝐵))
243, 18fpwwelem 10685 . . . . . . . . 9 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝑊(𝑊𝐵) ↔ ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))))
2523, 24mpbid 232 . . . . . . . 8 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)))
2625simprld 772 . . . . . . 7 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝑊𝐵) We 𝐵)
27 fvex 6919 . . . . . . . 8 (𝑊𝐵) ∈ V
28 weeq1 5672 . . . . . . . 8 (𝑟 = (𝑊𝐵) → (𝑟 We 𝐵 ↔ (𝑊𝐵) We 𝐵))
2927, 28spcev 3606 . . . . . . 7 ((𝑊𝐵) We 𝐵 → ∃𝑟 𝑟 We 𝐵)
3026, 29syl 17 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ∃𝑟 𝑟 We 𝐵)
31 ween 10075 . . . . . 6 (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵)
3230, 31sylibr 234 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ dom card)
3314, 32elind 4200 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ (𝒫 𝐴 ∩ dom card))
348simp2d 1144 . . . . . . . 8 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
3534pssssd 4100 . . . . . . 7 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
3635, 11sstrd 3994 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐴)
37 elpw2g 5333 . . . . . . 7 (𝐴𝑉 → (𝐶 ∈ 𝒫 𝐴𝐶𝐴))
3837adantr 480 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐶 ∈ 𝒫 𝐴𝐶𝐴))
3936, 38mpbird 257 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ 𝒫 𝐴)
40 ssnum 10079 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐶𝐵) → 𝐶 ∈ dom card)
4132, 35, 40syl2anc 584 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ dom card)
4239, 41elind 4200 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ (𝒫 𝐴 ∩ dom card))
43 f1fveq 7282 . . . 4 ((𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴 ∧ (𝐵 ∈ (𝒫 𝐴 ∩ dom card) ∧ 𝐶 ∈ (𝒫 𝐴 ∩ dom card))) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
4410, 33, 42, 43syl12anc 837 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
459, 44mpbid 232 . 2 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 = 𝐶)
4634pssned 4101 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
4746necomd 2996 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝐶)
4847neneqd 2945 . 2 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ¬ 𝐵 = 𝐶)
4945, 48pm2.65da 817 1 (𝐴𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wral 3061  cin 3950  wss 3951  wpss 3952  𝒫 cpw 4600  {csn 4626   cuni 4907   class class class wbr 5143  {copab 5205   We wwe 5636   × cxp 5683  ccnv 5684  dom cdm 5685  cima 5688  wf 6557  1-1wf1 6558  cfv 6561  cardccrd 9975
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-tp 4631  df-op 4633  df-uni 4908  df-int 4947  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-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-er 8745  df-en 8986  df-dom 8987  df-oi 9550  df-card 9979
This theorem is referenced by:  canthnum  10689
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