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Theorem canth4 10067
Description: An "effective" form of Cantor's theorem canth 7104. For any function 𝐹 from the powerset of 𝐴 to 𝐴, there are two definable sets 𝐵 and 𝐶 which witness non-injectivity of 𝐹. Corollary 1.3 of [KanamoriPincus] p. 416. (Contributed by Mario Carneiro, 18-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
canth4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
Distinct variable groups:   𝑥,𝑟,𝑦,𝐴   𝐵,𝑟,𝑥,𝑦   𝐷,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 canth4
StepHypRef Expression
1 eqid 2824 . . . . . . . 8 𝐵 = 𝐵
2 eqid 2824 . . . . . . . 8 (𝑊𝐵) = (𝑊𝐵)
31, 2pm3.2i 474 . . . . . . 7 (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))
4 canth4.1 . . . . . . . 8 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
5 elex 3498 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ V)
653ad2ant1 1130 . . . . . . . 8 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐴 ∈ V)
7 simpl2 1189 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝐹:𝐷𝐴)
8 simp3 1135 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝒫 𝐴 ∩ dom card) ⊆ 𝐷)
98sselda 3953 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝑥𝐷)
107, 9ffvelrnd 6843 . . . . . . . 8 (((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹𝑥) ∈ 𝐴)
11 canth4.2 . . . . . . . 8 𝐵 = dom 𝑊
124, 6, 10, 11fpwwe 10066 . . . . . . 7 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))))
133, 12mpbiri 261 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵))
1413simpld 498 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵𝑊(𝑊𝐵))
154, 6fpwwelem 10065 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝑊(𝑊𝐵) ↔ ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))))
1614, 15mpbid 235 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)))
1716simpld 498 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)))
1817simpld 498 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵𝐴)
19 canth4.3 . . . . 5 𝐶 = ((𝑊𝐵) “ {(𝐹𝐵)})
20 cnvimass 5936 . . . . 5 ((𝑊𝐵) “ {(𝐹𝐵)}) ⊆ dom (𝑊𝐵)
2119, 20eqsstri 3987 . . . 4 𝐶 ⊆ dom (𝑊𝐵)
2217simprd 499 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) ⊆ (𝐵 × 𝐵))
23 dmss 5758 . . . . . 6 ((𝑊𝐵) ⊆ (𝐵 × 𝐵) → dom (𝑊𝐵) ⊆ dom (𝐵 × 𝐵))
2422, 23syl 17 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊𝐵) ⊆ dom (𝐵 × 𝐵))
25 dmxpid 5787 . . . . 5 dom (𝐵 × 𝐵) = 𝐵
2624, 25sseqtrdi 4003 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊𝐵) ⊆ 𝐵)
2721, 26sstrid 3964 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶𝐵)
2813simprd 499 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐵) ∈ 𝐵)
2916simprd 499 . . . . . . 7 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))
3029simpld 498 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) We 𝐵)
31 weso 5533 . . . . . 6 ((𝑊𝐵) We 𝐵 → (𝑊𝐵) Or 𝐵)
3230, 31syl 17 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) Or 𝐵)
33 sonr 5483 . . . . 5 (((𝑊𝐵) Or 𝐵 ∧ (𝐹𝐵) ∈ 𝐵) → ¬ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3432, 28, 33syl2anc 587 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3519eleq2i 2907 . . . . 5 ((𝐹𝐵) ∈ 𝐶 ↔ (𝐹𝐵) ∈ ((𝑊𝐵) “ {(𝐹𝐵)}))
36 fvex 6674 . . . . . 6 (𝐹𝐵) ∈ V
3736eliniseg 5945 . . . . . 6 ((𝐹𝐵) ∈ V → ((𝐹𝐵) ∈ ((𝑊𝐵) “ {(𝐹𝐵)}) ↔ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵)))
3836, 37ax-mp 5 . . . . 5 ((𝐹𝐵) ∈ ((𝑊𝐵) “ {(𝐹𝐵)}) ↔ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3935, 38bitri 278 . . . 4 ((𝐹𝐵) ∈ 𝐶 ↔ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
4034, 39sylnibr 332 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹𝐵) ∈ 𝐶)
4127, 28, 40ssnelpssd 4075 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶𝐵)
42 sneq 4560 . . . . . . . 8 (𝑦 = (𝐹𝐵) → {𝑦} = {(𝐹𝐵)})
4342imaeq2d 5916 . . . . . . 7 (𝑦 = (𝐹𝐵) → ((𝑊𝐵) “ {𝑦}) = ((𝑊𝐵) “ {(𝐹𝐵)}))
4443, 19syl6eqr 2877 . . . . . 6 (𝑦 = (𝐹𝐵) → ((𝑊𝐵) “ {𝑦}) = 𝐶)
4544fveq2d 6665 . . . . 5 (𝑦 = (𝐹𝐵) → (𝐹‘((𝑊𝐵) “ {𝑦})) = (𝐹𝐶))
46 id 22 . . . . 5 (𝑦 = (𝐹𝐵) → 𝑦 = (𝐹𝐵))
4745, 46eqeq12d 2840 . . . 4 (𝑦 = (𝐹𝐵) → ((𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦 ↔ (𝐹𝐶) = (𝐹𝐵)))
4829simprd 499 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)
4947, 48, 28rspcdva 3611 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐶) = (𝐹𝐵))
5049eqcomd 2830 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐵) = (𝐹𝐶))
5118, 41, 503jca 1125 1 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  Vcvv 3480  cin 3918  wss 3919  wpss 3920  𝒫 cpw 4522  {csn 4550   cuni 4824   class class class wbr 5052  {copab 5114   Or wor 5460   We wwe 5500   × cxp 5540  ccnv 5541  dom cdm 5542  cima 5545  wf 6339  cfv 6343  cardccrd 9361
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-1st 7684  df-wrecs 7943  df-recs 8004  df-en 8506  df-oi 8971  df-card 9365
This theorem is referenced by:  canthnumlem  10068  canthp1lem2  10073
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