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Theorem canth4 9507
Description: An "effective" form of Cantor's theorem canth 6648. 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 2651 . . . . . . . 8 𝐵 = 𝐵
2 eqid 2651 . . . . . . . 8 (𝑊𝐵) = (𝑊𝐵)
31, 2pm3.2i 470 . . . . . . 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 3243 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ V)
653ad2ant1 1102 . . . . . . . 8 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐴 ∈ V)
7 simpl2 1085 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝐹:𝐷𝐴)
8 simp3 1083 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝒫 𝐴 ∩ dom card) ⊆ 𝐷)
98sselda 3636 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝑥𝐷)
107, 9ffvelrnd 6400 . . . . . . . 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 9506 . . . . . . 7 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))))
133, 12mpbiri 248 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵))
1413simpld 474 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵𝑊(𝑊𝐵))
154, 6fpwwelem 9505 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝑊(𝑊𝐵) ↔ ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))))
1614, 15mpbid 222 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)))
1716simpld 474 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)))
1817simpld 474 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵𝐴)
19 canth4.3 . . . . 5 𝐶 = ((𝑊𝐵) “ {(𝐹𝐵)})
20 cnvimass 5520 . . . . 5 ((𝑊𝐵) “ {(𝐹𝐵)}) ⊆ dom (𝑊𝐵)
2119, 20eqsstri 3668 . . . 4 𝐶 ⊆ dom (𝑊𝐵)
2217simprd 478 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) ⊆ (𝐵 × 𝐵))
23 dmss 5355 . . . . . 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 5377 . . . . 5 dom (𝐵 × 𝐵) = 𝐵
2624, 25syl6sseq 3684 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊𝐵) ⊆ 𝐵)
2721, 26syl5ss 3647 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶𝐵)
2813simprd 478 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐵) ∈ 𝐵)
2916simprd 478 . . . . . . 7 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))
3029simpld 474 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) We 𝐵)
31 weso 5134 . . . . . 6 ((𝑊𝐵) We 𝐵 → (𝑊𝐵) Or 𝐵)
3230, 31syl 17 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) Or 𝐵)
33 sonr 5085 . . . . 5 (((𝑊𝐵) Or 𝐵 ∧ (𝐹𝐵) ∈ 𝐵) → ¬ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3432, 28, 33syl2anc 694 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3519eleq2i 2722 . . . . 5 ((𝐹𝐵) ∈ 𝐶 ↔ (𝐹𝐵) ∈ ((𝑊𝐵) “ {(𝐹𝐵)}))
36 fvex 6239 . . . . . 6 (𝐹𝐵) ∈ V
3736eliniseg 5529 . . . . . 6 ((𝐹𝐵) ∈ V → ((𝐹𝐵) ∈ ((𝑊𝐵) “ {(𝐹𝐵)}) ↔ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵)))
3836, 37ax-mp 5 . . . . 5 ((𝐹𝐵) ∈ ((𝑊𝐵) “ {(𝐹𝐵)}) ↔ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3935, 38bitri 264 . . . 4 ((𝐹𝐵) ∈ 𝐶 ↔ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
4034, 39sylnibr 318 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹𝐵) ∈ 𝐶)
4127, 28, 40ssnelpssd 3752 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶𝐵)
4229simprd 478 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)
43 sneq 4220 . . . . . . . . 9 (𝑦 = (𝐹𝐵) → {𝑦} = {(𝐹𝐵)})
4443imaeq2d 5501 . . . . . . . 8 (𝑦 = (𝐹𝐵) → ((𝑊𝐵) “ {𝑦}) = ((𝑊𝐵) “ {(𝐹𝐵)}))
4544, 19syl6eqr 2703 . . . . . . 7 (𝑦 = (𝐹𝐵) → ((𝑊𝐵) “ {𝑦}) = 𝐶)
4645fveq2d 6233 . . . . . 6 (𝑦 = (𝐹𝐵) → (𝐹‘((𝑊𝐵) “ {𝑦})) = (𝐹𝐶))
47 id 22 . . . . . 6 (𝑦 = (𝐹𝐵) → 𝑦 = (𝐹𝐵))
4846, 47eqeq12d 2666 . . . . 5 (𝑦 = (𝐹𝐵) → ((𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦 ↔ (𝐹𝐶) = (𝐹𝐵)))
4948rspcv 3336 . . . 4 ((𝐹𝐵) ∈ 𝐵 → (∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦 → (𝐹𝐶) = (𝐹𝐵)))
5028, 42, 49sylc 65 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐶) = (𝐹𝐵))
5150eqcomd 2657 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐵) = (𝐹𝐶))
5218, 41, 513jca 1261 1 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cin 3606  wss 3607  wpss 3608  𝒫 cpw 4191  {csn 4210   cuni 4468   class class class wbr 4685  {copab 4745   Or wor 5063   We wwe 5101   × cxp 5141  ccnv 5142  dom cdm 5143  cima 5146  wf 5922  cfv 5926  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-1st 7210  df-wrecs 7452  df-recs 7513  df-en 7998  df-oi 8456  df-card 8803
This theorem is referenced by:  canthnumlem  9508  canthp1lem2  9513
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