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Theorem canth4 10642
Description: An "effective" form of Cantor's theorem canth 7362. 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 2733 . . . . . . . 8 𝐵 = 𝐵
2 eqid 2733 . . . . . . . 8 (𝑊𝐵) = (𝑊𝐵)
31, 2pm3.2i 472 . . . . . . 7 (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))
4 canth4.1 . . . . . . . 8 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 (𝐹‘(𝑟 “ {𝑦})) = 𝑦))}
5 simp1 1137 . . . . . . . 8 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐴𝑉)
6 simpl2 1193 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝐹:𝐷𝐴)
7 simp3 1139 . . . . . . . . . 10 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝒫 𝐴 ∩ dom card) ⊆ 𝐷)
87sselda 3983 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝑥𝐷)
96, 8ffvelcdmd 7088 . . . . . . . 8 (((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹𝑥) ∈ 𝐴)
10 canth4.2 . . . . . . . 8 𝐵 = dom 𝑊
114, 5, 9, 10fpwwe 10641 . . . . . . 7 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊𝐵) = (𝑊𝐵))))
123, 11mpbiri 258 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝑊(𝑊𝐵) ∧ (𝐹𝐵) ∈ 𝐵))
1312simpld 496 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵𝑊(𝑊𝐵))
144, 5fpwwelem 10640 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝑊(𝑊𝐵) ↔ ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))))
1513, 14mpbid 231 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)))
1615simpld 496 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝐴 ∧ (𝑊𝐵) ⊆ (𝐵 × 𝐵)))
1716simpld 496 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵𝐴)
18 canth4.3 . . . . 5 𝐶 = ((𝑊𝐵) “ {(𝐹𝐵)})
19 cnvimass 6081 . . . . 5 ((𝑊𝐵) “ {(𝐹𝐵)}) ⊆ dom (𝑊𝐵)
2018, 19eqsstri 4017 . . . 4 𝐶 ⊆ dom (𝑊𝐵)
2116simprd 497 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) ⊆ (𝐵 × 𝐵))
22 dmss 5903 . . . . . 6 ((𝑊𝐵) ⊆ (𝐵 × 𝐵) → dom (𝑊𝐵) ⊆ dom (𝐵 × 𝐵))
2321, 22syl 17 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊𝐵) ⊆ dom (𝐵 × 𝐵))
24 dmxpid 5930 . . . . 5 dom (𝐵 × 𝐵) = 𝐵
2523, 24sseqtrdi 4033 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊𝐵) ⊆ 𝐵)
2620, 25sstrid 3994 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶𝐵)
2712simprd 497 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐵) ∈ 𝐵)
2815simprd 497 . . . . . . 7 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝑊𝐵) We 𝐵 ∧ ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦))
2928simpld 496 . . . . . 6 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) We 𝐵)
30 weso 5668 . . . . . 6 ((𝑊𝐵) We 𝐵 → (𝑊𝐵) Or 𝐵)
3129, 30syl 17 . . . . 5 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊𝐵) Or 𝐵)
32 sonr 5612 . . . . 5 (((𝑊𝐵) Or 𝐵 ∧ (𝐹𝐵) ∈ 𝐵) → ¬ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3331, 27, 32syl2anc 585 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3418eleq2i 2826 . . . . 5 ((𝐹𝐵) ∈ 𝐶 ↔ (𝐹𝐵) ∈ ((𝑊𝐵) “ {(𝐹𝐵)}))
35 fvex 6905 . . . . . 6 (𝐹𝐵) ∈ V
3635eliniseg 6094 . . . . . 6 ((𝐹𝐵) ∈ V → ((𝐹𝐵) ∈ ((𝑊𝐵) “ {(𝐹𝐵)}) ↔ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵)))
3735, 36ax-mp 5 . . . . 5 ((𝐹𝐵) ∈ ((𝑊𝐵) “ {(𝐹𝐵)}) ↔ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3834, 37bitri 275 . . . 4 ((𝐹𝐵) ∈ 𝐶 ↔ (𝐹𝐵)(𝑊𝐵)(𝐹𝐵))
3933, 38sylnibr 329 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹𝐵) ∈ 𝐶)
4026, 27, 39ssnelpssd 4113 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶𝐵)
41 sneq 4639 . . . . . . . 8 (𝑦 = (𝐹𝐵) → {𝑦} = {(𝐹𝐵)})
4241imaeq2d 6060 . . . . . . 7 (𝑦 = (𝐹𝐵) → ((𝑊𝐵) “ {𝑦}) = ((𝑊𝐵) “ {(𝐹𝐵)}))
4342, 18eqtr4di 2791 . . . . . 6 (𝑦 = (𝐹𝐵) → ((𝑊𝐵) “ {𝑦}) = 𝐶)
4443fveq2d 6896 . . . . 5 (𝑦 = (𝐹𝐵) → (𝐹‘((𝑊𝐵) “ {𝑦})) = (𝐹𝐶))
45 id 22 . . . . 5 (𝑦 = (𝐹𝐵) → 𝑦 = (𝐹𝐵))
4644, 45eqeq12d 2749 . . . 4 (𝑦 = (𝐹𝐵) → ((𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦 ↔ (𝐹𝐶) = (𝐹𝐵)))
4728simprd 497 . . . 4 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ∀𝑦𝐵 (𝐹‘((𝑊𝐵) “ {𝑦})) = 𝑦)
4846, 47, 27rspcdva 3614 . . 3 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐶) = (𝐹𝐵))
4948eqcomd 2739 . 2 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹𝐵) = (𝐹𝐶))
5017, 40, 493jca 1129 1 ((𝐴𝑉𝐹:𝐷𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝐴𝐶𝐵 ∧ (𝐹𝐵) = (𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  cin 3948  wss 3949  wpss 3950  𝒫 cpw 4603  {csn 4629   cuni 4909   class class class wbr 5149  {copab 5211   Or wor 5588   We wwe 5631   × cxp 5675  ccnv 5676  dom cdm 5677  cima 5680  wf 6540  cfv 6544  cardccrd 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-en 8940  df-oi 9505  df-card 9934
This theorem is referenced by:  canthnumlem  10643  canthp1lem2  10648
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