MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  canth4 Structured version   Visualization version   GIF version

Theorem canth4 9321
Description: An "effective" form of Cantor's theorem canth 6482. 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 2605 . . . . . . . 8 𝐵 = 𝐵
2 eqid 2605 . . . . . . . 8 (𝑊𝐵) = (𝑊𝐵)
31, 2pm3.2i 469 . . . . . . 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 3180 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ V)
653ad2ant1 1074 . . . . . . . 8 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐴 ∈ V)
7 simpl2 1057 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝐹:𝐷𝐴)
8 simp3 1055 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝒫 𝐴 ∩ dom card) ⊆ 𝐷)
98sselda 3563 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝑥𝐷)
107, 9ffvelrnd 6249 . . . . . . . 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 9320 . . . . . . 7 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))))
133, 12mpbiri 246 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵))
1413simpld 473 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵𝑊(𝑊𝐵))
154, 6fpwwelem 9319 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝑊(𝑊𝐵) ↔ ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))))
1614, 15mpbid 220 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)))
1716simpld 473 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)))
1817simpld 473 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵𝐴)
19 canth4.3 . . . . 5 𝐶 = ((𝑊𝐵) “ {(𝐹𝐵)})
20 cnvimass 5387 . . . . 5 ((𝑊𝐵) “ {(𝐹𝐵)}) ⊆ dom (𝑊𝐵)
2119, 20eqsstri 3593 . . . 4 𝐶 ⊆ dom (𝑊𝐵)
2217simprd 477 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) ⊆ (𝐵 × 𝐵))
23 dmss 5228 . . . . . 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 5249 . . . . 5 dom (𝐵 × 𝐵) = 𝐵
2624, 25syl6sseq 3609 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊𝐵) ⊆ 𝐵)
2721, 26syl5ss 3574 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶𝐵)
2813simprd 477 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐵) ∈ 𝐵)
2916simprd 477 . . . . . . 7 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))
3029simpld 473 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) We 𝐵)
31 weso 5015 . . . . . 6 ((𝑊𝐵) We 𝐵 → (𝑊𝐵) Or 𝐵)
3230, 31syl 17 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) Or 𝐵)
33 sonr 4966 . . . . 5 (((𝑊𝐵) Or 𝐵 ∧ (𝐹𝐵) ∈ 𝐵) → ¬ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3432, 28, 33syl2anc 690 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3519eleq2i 2675 . . . . 5 ((𝐹𝐵) ∈ 𝐶 ↔ (𝐹𝐵) ∈ ((𝑊𝐵) “ {(𝐹𝐵)}))
36 fvex 6094 . . . . . 6 (𝐹𝐵) ∈ V
3736eliniseg 5396 . . . . . 6 ((𝐹𝐵) ∈ V → ((𝐹𝐵) ∈ ((𝑊𝐵) “ {(𝐹𝐵)}) ↔ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵)))
3836, 37ax-mp 5 . . . . 5 ((𝐹𝐵) ∈ ((𝑊𝐵) “ {(𝐹𝐵)}) ↔ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3935, 38bitri 262 . . . 4 ((𝐹𝐵) ∈ 𝐶 ↔ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
4034, 39sylnibr 317 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹𝐵) ∈ 𝐶)
4127, 28, 40ssnelpssd 3676 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶𝐵)
4229simprd 477 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)
43 sneq 4130 . . . . . . . . 9 (𝑦 = (𝐹𝐵) → {𝑦} = {(𝐹𝐵)})
4443imaeq2d 5368 . . . . . . . 8 (𝑦 = (𝐹𝐵) → ((𝑊𝐵) “ {𝑦}) = ((𝑊𝐵) “ {(𝐹𝐵)}))
4544, 19syl6eqr 2657 . . . . . . 7 (𝑦 = (𝐹𝐵) → ((𝑊𝐵) “ {𝑦}) = 𝐶)
4645fveq2d 6088 . . . . . 6 (𝑦 = (𝐹𝐵) → (𝐹‘((𝑊𝐵) “ {𝑦})) = (𝐹𝐶))
47 id 22 . . . . . 6 (𝑦 = (𝐹𝐵) → 𝑦 = (𝐹𝐵))
4846, 47eqeq12d 2620 . . . . 5 (𝑦 = (𝐹𝐵) → ((𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦 ↔ (𝐹𝐶) = (𝐹𝐵)))
4948rspcv 3273 . . . 4 ((𝐹𝐵) ∈ 𝐵 → (∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦 → (𝐹𝐶) = (𝐹𝐵)))
5028, 42, 49sylc 62 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐶) = (𝐹𝐵))
5150eqcomd 2611 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐵) = (𝐹𝐶))
5218, 41, 513jca 1234 1 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wral 2891  Vcvv 3168  cin 3534  wss 3535  wpss 3536  𝒫 cpw 4103  {csn 4120   cuni 4362   class class class wbr 4573  {copab 4632   Or wor 4944   We wwe 4982   × cxp 5022  ccnv 5023  dom cdm 5024  cima 5027  wf 5782  cfv 5786  cardccrd 8617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-ov 6526  df-1st 7032  df-wrecs 7267  df-recs 7328  df-en 7815  df-oi 8271  df-card 8621
This theorem is referenced by:  canthnumlem  9322  canthp1lem2  9327
  Copyright terms: Public domain W3C validator