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Theorem canth4 9454
Description: An "effective" form of Cantor's theorem canth 6593. 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 2620 . . . . . . . 8 𝐵 = 𝐵
2 eqid 2620 . . . . . . . 8 (𝑊𝐵) = (𝑊𝐵)
31, 2pm3.2i 471 . . . . . . 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 3207 . . . . . . . . 9 (𝐴𝑉𝐴 ∈ V)
653ad2ant1 1080 . . . . . . . 8 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐴 ∈ V)
7 simpl2 1063 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝐹:𝐷𝐴)
8 simp3 1061 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝒫 𝐴 ∩ dom card) ⊆ 𝐷)
98sselda 3595 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝑥𝐷)
107, 9ffvelrnd 6346 . . . . . . . 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 9453 . . . . . . 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 475 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵𝑊(𝑊𝐵))
154, 6fpwwelem 9452 . . . . 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 475 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)))
1817simpld 475 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵𝐴)
19 canth4.3 . . . . 5 𝐶 = ((𝑊𝐵) “ {(𝐹𝐵)})
20 cnvimass 5473 . . . . 5 ((𝑊𝐵) “ {(𝐹𝐵)}) ⊆ dom (𝑊𝐵)
2119, 20eqsstri 3627 . . . 4 𝐶 ⊆ dom (𝑊𝐵)
2217simprd 479 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) ⊆ (𝐵 × 𝐵))
23 dmss 5312 . . . . . 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 5334 . . . . 5 dom (𝐵 × 𝐵) = 𝐵
2624, 25syl6sseq 3643 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊𝐵) ⊆ 𝐵)
2721, 26syl5ss 3606 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶𝐵)
2813simprd 479 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐵) ∈ 𝐵)
2916simprd 479 . . . . . . 7 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))
3029simpld 475 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) We 𝐵)
31 weso 5095 . . . . . 6 ((𝑊𝐵) We 𝐵 → (𝑊𝐵) Or 𝐵)
3230, 31syl 17 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) Or 𝐵)
33 sonr 5046 . . . . 5 (((𝑊𝐵) Or 𝐵 ∧ (𝐹𝐵) ∈ 𝐵) → ¬ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3432, 28, 33syl2anc 692 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3519eleq2i 2691 . . . . 5 ((𝐹𝐵) ∈ 𝐶 ↔ (𝐹𝐵) ∈ ((𝑊𝐵) “ {(𝐹𝐵)}))
36 fvex 6188 . . . . . 6 (𝐹𝐵) ∈ V
3736eliniseg 5482 . . . . . 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 319 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹𝐵) ∈ 𝐶)
4127, 28, 40ssnelpssd 3711 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶𝐵)
4229simprd 479 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)
43 sneq 4178 . . . . . . . . 9 (𝑦 = (𝐹𝐵) → {𝑦} = {(𝐹𝐵)})
4443imaeq2d 5454 . . . . . . . 8 (𝑦 = (𝐹𝐵) → ((𝑊𝐵) “ {𝑦}) = ((𝑊𝐵) “ {(𝐹𝐵)}))
4544, 19syl6eqr 2672 . . . . . . 7 (𝑦 = (𝐹𝐵) → ((𝑊𝐵) “ {𝑦}) = 𝐶)
4645fveq2d 6182 . . . . . 6 (𝑦 = (𝐹𝐵) → (𝐹‘((𝑊𝐵) “ {𝑦})) = (𝐹𝐶))
47 id 22 . . . . . 6 (𝑦 = (𝐹𝐵) → 𝑦 = (𝐹𝐵))
4846, 47eqeq12d 2635 . . . . 5 (𝑦 = (𝐹𝐵) → ((𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦 ↔ (𝐹𝐶) = (𝐹𝐵)))
4948rspcv 3300 . . . 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 2626 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐵) = (𝐹𝐶))
5218, 41, 513jca 1240 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 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  cin 3566  wss 3567  wpss 3568  𝒫 cpw 4149  {csn 4168   cuni 4427   class class class wbr 4644  {copab 4703   Or wor 5024   We wwe 5062   × cxp 5102  ccnv 5103  dom cdm 5104  cima 5107  wf 5872  cfv 5876  cardccrd 8746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-1st 7153  df-wrecs 7392  df-recs 7453  df-en 7941  df-oi 8400  df-card 8750
This theorem is referenced by:  canthnumlem  9455  canthp1lem2  9460
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