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Theorem canthnumlem 9916
Description: Lemma for canthnum 9917. (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 6443 . . . . 5 (𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2 ssid 3910 . . . . . 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 9915 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ (𝒫 𝐴 ∩ dom card)) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
72, 6mp3an3 1442 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
81, 7sylan2 592 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
98simp3d 1137 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐹𝐵) = (𝐹𝐶))
10 simpr 485 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
118simp1d 1135 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝐴)
12 elpw2g 5138 . . . . . . 7 (𝐴𝑉 → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1312adantr 481 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵 ∈ 𝒫 𝐴𝐵𝐴))
1411, 13mpbird 258 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ 𝒫 𝐴)
15 eqid 2795 . . . . . . . . . . . 12 𝐵 = 𝐵
16 eqid 2795 . . . . . . . . . . . 12 (𝑊𝐵) = (𝑊𝐵)
1715, 16pm3.2i 471 . . . . . . . . . . 11 (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))
18 elex 3455 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 ∈ V)
1918adantr 481 . . . . . . . . . . . 12 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐴 ∈ V)
2010, 1syl 17 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2120ffvelrnda 6716 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹𝑥) ∈ 𝐴)
223, 19, 21, 4fpwwe 9914 . . . . . . . . . . 11 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))))
2317, 22mpbiri 259 . . . . . . . . . 10 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵))
2423simpld 495 . . . . . . . . 9 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝑊(𝑊𝐵))
253, 19fpwwelem 9913 . . . . . . . . 9 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐵𝑊(𝑊𝐵) ↔ ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))))
2624, 25mpbid 233 . . . . . . . 8 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)))
2726simprld 768 . . . . . . 7 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝑊𝐵) We 𝐵)
28 fvex 6551 . . . . . . . 8 (𝑊𝐵) ∈ V
29 weeq1 5431 . . . . . . . 8 (𝑟 = (𝑊𝐵) → (𝑟 We 𝐵 ↔ (𝑊𝐵) We 𝐵))
3028, 29spcev 3549 . . . . . . 7 ((𝑊𝐵) We 𝐵 → ∃𝑟 𝑟 We 𝐵)
3127, 30syl 17 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ∃𝑟 𝑟 We 𝐵)
32 ween 9307 . . . . . 6 (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵)
3331, 32sylibr 235 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ dom card)
3414, 33elind 4092 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 ∈ (𝒫 𝐴 ∩ dom card))
358simp2d 1136 . . . . . . . 8 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
3635pssssd 3995 . . . . . . 7 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
3736, 11sstrd 3899 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐴)
38 elpw2g 5138 . . . . . . 7 (𝐴𝑉 → (𝐶 ∈ 𝒫 𝐴𝐶𝐴))
3938adantr 481 . . . . . 6 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → (𝐶 ∈ 𝒫 𝐴𝐶𝐴))
4037, 39mpbird 258 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ 𝒫 𝐴)
41 ssnum 9311 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐶𝐵) → 𝐶 ∈ dom card)
4233, 36, 41syl2anc 584 . . . . 5 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ dom card)
4340, 42elind 4092 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶 ∈ (𝒫 𝐴 ∩ dom card))
44 f1fveq 6885 . . . 4 ((𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴 ∧ (𝐵 ∈ (𝒫 𝐴 ∩ dom card) ∧ 𝐶 ∈ (𝒫 𝐴 ∩ dom card))) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
4510, 34, 43, 44syl12anc 833 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ((𝐹𝐵) = (𝐹𝐶) ↔ 𝐵 = 𝐶))
469, 45mpbid 233 . 2 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵 = 𝐶)
4735pssned 3996 . . . 4 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐶𝐵)
4847necomd 3039 . . 3 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → 𝐵𝐶)
4948neneqd 2989 . 2 ((𝐴𝑉𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴) → ¬ 𝐵 = 𝐶)
5046, 49pm2.65da 813 1 (𝐴𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wex 1761  wcel 2081  wral 3105  Vcvv 3437  cin 3858  wss 3859  wpss 3860  𝒫 cpw 4453  {csn 4472   cuni 4745   class class class wbr 4962  {copab 5024   We wwe 5401   × cxp 5441  ccnv 5442  dom cdm 5443  cima 5446  wf 6221  1-1wf1 6222  cfv 6225  cardccrd 9210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-1st 7545  df-wrecs 7798  df-recs 7860  df-er 8139  df-en 8358  df-dom 8359  df-oi 8820  df-card 9214
This theorem is referenced by:  canthnum  9917
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