Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  canthnumlem Structured version   Visualization version   GIF version

Theorem canthnumlem 10077
 Description: Lemma for canthnum 10078. (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 6557 . . . . 5 (𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2 ssid 3939 . . . . . 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 10076 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ (𝒫 𝐴 ∩ dom card)) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
72, 6mp3an3 1447 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
81, 7sylan2 595 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
98simp3d 1141 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐹𝐵) = (𝐹𝐶))
10 simpr 488 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
118simp1d 1139 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝐴)
12 elpw2g 5215 . . . . . . 7 (𝐴𝑉 → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1312adantr 484 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1411, 13mpbird 260 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ 𝒫 𝐴)
15 eqid 2798 . . . . . . . . . . . 12 𝐵 = 𝐵
16 eqid 2798 . . . . . . . . . . . 12 (𝑊𝐵) = (𝑊𝐵)
1715, 16pm3.2i 474 . . . . . . . . . . 11 (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))
18 simpl 486 . . . . . . . . . . . 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)
2019ffvelrnda 6838 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹𝑥) ∈ 𝐴)
213, 18, 20, 4fpwwe 10075 . . . . . . . . . . 11 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))))
2217, 21mpbiri 261 . . . . . . . . . 10 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵))
2322simpld 498 . . . . . . . . 9 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝑊(𝑊𝐵))
243, 18fpwwelem 10074 . . . . . . . . 9 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝑊(𝑊𝐵) ↔ ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))))
2523, 24mpbid 235 . . . . . . . 8 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)))
2625simprld 771 . . . . . . 7 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝑊𝐵) We 𝐵)
27 fvex 6668 . . . . . . . 8 (𝑊𝐵) ∈ V
28 weeq1 5511 . . . . . . . 8 (𝑟 = (𝑊𝐵) → (𝑟 We 𝐵 ↔ (𝑊𝐵) We 𝐵))
2927, 28spcev 3556 . . . . . . 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 9464 . . . . . 6 (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵)
3230, 31sylibr 237 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ dom card)
3314, 32elind 4124 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ (𝒫 𝐴 ∩ dom card))
348simp2d 1140 . . . . . . . 8 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
3534pssssd 4028 . . . . . . 7 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
3635, 11sstrd 3927 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐴)
37 elpw2g 5215 . . . . . . 7 (𝐴𝑉 → (𝐶 ∈ 𝒫 𝐴𝐶𝐴))
3837adantr 484 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐶 ∈ 𝒫 𝐴𝐶𝐴))
3936, 38mpbird 260 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ 𝒫 𝐴)
40 ssnum 9468 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐶𝐵) → 𝐶 ∈ dom card)
4132, 35, 40syl2anc 587 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ dom card)
4239, 41elind 4124 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ (𝒫 𝐴 ∩ dom card))
43 f1fveq 7008 . . . 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 835 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
459, 44mpbid 235 . 2 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 = 𝐶)
4634pssned 4029 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
4746necomd 3042 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝐶)
4847neneqd 2992 . 2 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ¬ 𝐵 = 𝐶)
4945, 48pm2.65da 816 1 (𝐴𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3106   ∩ cin 3882   ⊆ wss 3883   ⊊ wpss 3884  𝒫 cpw 4500  {csn 4528  ∪ cuni 4804   class class class wbr 5034  {copab 5096   We wwe 5481   × cxp 5521  ◡ccnv 5522  dom cdm 5523   “ cima 5526  ⟶wf 6328  –1-1→wf1 6329  ‘cfv 6332  cardccrd 9366 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-isom 6341  df-riota 7103  df-ov 7148  df-1st 7684  df-wrecs 7948  df-recs 8009  df-er 8290  df-en 8511  df-dom 8512  df-oi 8976  df-card 9370 This theorem is referenced by:  canthnum  10078
 Copyright terms: Public domain W3C validator