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Theorem canthnumlem 9417
Description: Lemma for canthnum 9418. (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 6060 . . . . 5 (𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2 ssid 3605 . . . . . 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 9416 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ (𝒫 𝐴 ∩ dom card)) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
72, 6mp3an3 1410 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
81, 7sylan2 491 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
98simp3d 1073 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐹𝐵) = (𝐹𝐶))
10 simpr 477 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
118simp1d 1071 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝐴)
12 elpw2g 4789 . . . . . . 7 (𝐴𝑉 → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1312adantr 481 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1411, 13mpbird 247 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ 𝒫 𝐴)
15 eqid 2621 . . . . . . . . . . . . 13 𝐵 = 𝐵
16 eqid 2621 . . . . . . . . . . . . 13 (𝑊𝐵) = (𝑊𝐵)
1715, 16pm3.2i 471 . . . . . . . . . . . 12 (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))
18 elex 3198 . . . . . . . . . . . . . 14 (𝐴𝑉𝐴 ∈ V)
1918adantr 481 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐴 ∈ V)
2010, 1syl 17 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2120ffvelrnda 6317 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹𝑥) ∈ 𝐴)
223, 19, 21, 4fpwwe 9415 . . . . . . . . . . . 12 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))))
2317, 22mpbiri 248 . . . . . . . . . . 11 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵))
2423simpld 475 . . . . . . . . . 10 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝑊(𝑊𝐵))
253, 19fpwwelem 9414 . . . . . . . . . 10 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝑊(𝑊𝐵) ↔ ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))))
2624, 25mpbid 222 . . . . . . . . 9 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)))
2726simprd 479 . . . . . . . 8 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))
2827simpld 475 . . . . . . 7 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝑊𝐵) We 𝐵)
29 fvex 6160 . . . . . . . 8 (𝑊𝐵) ∈ V
30 weeq1 5064 . . . . . . . 8 (𝑟 = (𝑊𝐵) → (𝑟 We 𝐵 ↔ (𝑊𝐵) We 𝐵))
3129, 30spcev 3286 . . . . . . 7 ((𝑊𝐵) We 𝐵 → ∃𝑟 𝑟 We 𝐵)
3228, 31syl 17 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ∃𝑟 𝑟 We 𝐵)
33 ween 8805 . . . . . 6 (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵)
3432, 33sylibr 224 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ dom card)
3514, 34elind 3778 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ (𝒫 𝐴 ∩ dom card))
368simp2d 1072 . . . . . . . 8 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
3736pssssd 3684 . . . . . . 7 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
3837, 11sstrd 3594 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐴)
39 elpw2g 4789 . . . . . . 7 (𝐴𝑉 → (𝐶 ∈ 𝒫 𝐴𝐶𝐴))
4039adantr 481 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐶 ∈ 𝒫 𝐴𝐶𝐴))
4138, 40mpbird 247 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ 𝒫 𝐴)
42 ssnum 8809 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐶𝐵) → 𝐶 ∈ dom card)
4334, 37, 42syl2anc 692 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ dom card)
4441, 43elind 3778 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ (𝒫 𝐴 ∩ dom card))
45 f1fveq 6476 . . . 4 ((𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴 ∧ (𝐵 ∈ (𝒫 𝐴 ∩ dom card) ∧ 𝐶 ∈ (𝒫 𝐴 ∩ dom card))) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
4610, 35, 44, 45syl12anc 1321 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
479, 46mpbid 222 . 2 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 = 𝐶)
4836pssned 3685 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
4948necomd 2845 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝐶)
5049neneqd 2795 . 2 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ¬ 𝐵 = 𝐶)
5147, 50pm2.65da 599 1 (𝐴𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wral 2907  Vcvv 3186  cin 3555  wss 3556  wpss 3557  𝒫 cpw 4132  {csn 4150   cuni 4404   class class class wbr 4615  {copab 4674   We wwe 5034   × cxp 5074  ccnv 5075  dom cdm 5076  cima 5079  wf 5845  1-1wf1 5846  cfv 5849  cardccrd 8708
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 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-se 5036  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-isom 5858  df-riota 6568  df-ov 6610  df-1st 7116  df-wrecs 7355  df-recs 7416  df-er 7690  df-en 7903  df-dom 7904  df-oi 8362  df-card 8712
This theorem is referenced by:  canthnum  9418
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