Theorem canth4 10334

Description: An "effective" form of Cantor's theorem canth 7209. 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⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}
canth4.2	⊢ 𝐵 = ∪ dom 𝑊
canth4.3	⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)})

Assertion

Ref	Expression
canth4	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶)))

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

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . . . . 8 ⊢ 𝐵 = 𝐵
2		eqid 2738	. . . . . . . 8 ⊢ (𝑊‘𝐵) = (𝑊‘𝐵)
3	1, 2	pm3.2i 470	. . . . . . 7 ⊢ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))
4		canth4.1	. . . . . . . 8 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}
5		simp1 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐴 ∈ 𝑉)
6		simpl2 1190	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝐹:𝐷⟶𝐴)
7		simp3 1136	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝒫 𝐴 ∩ dom card) ⊆ 𝐷)
8	7	sselda 3917	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝑥 ∈ 𝐷)
9	6, 8	ffvelrnd 6944	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹‘𝑥) ∈ 𝐴)
10		canth4.2	. . . . . . . 8 ⊢ 𝐵 = ∪ dom 𝑊
11	4, 5, 9, 10	fpwwe 10333	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝐵𝑊(𝑊‘𝐵) ∧ (𝐹‘𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))))
12	3, 11	mpbiri 257	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝑊(𝑊‘𝐵) ∧ (𝐹‘𝐵) ∈ 𝐵))
13	12	simpld 494	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵𝑊(𝑊‘𝐵))
14	4, 5	fpwwelem 10332	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝑊(𝑊‘𝐵) ↔ ((𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦))))
15	13, 14	mpbid 231	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦)))
16	15	simpld 494	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)))
17	16	simpld 494	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵 ⊆ 𝐴)
18		canth4.3	. . . . 5 ⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)})
19		cnvimass 5978	. . . . 5 ⊢ (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}) ⊆ dom (𝑊‘𝐵)
20	18, 19	eqsstri 3951	. . . 4 ⊢ 𝐶 ⊆ dom (𝑊‘𝐵)
21	16	simprd 495	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊‘𝐵) ⊆ (𝐵 × 𝐵))
22		dmss 5800	. . . . . 6 ⊢ ((𝑊‘𝐵) ⊆ (𝐵 × 𝐵) → dom (𝑊‘𝐵) ⊆ dom (𝐵 × 𝐵))
23	21, 22	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊‘𝐵) ⊆ dom (𝐵 × 𝐵))
24		dmxpid 5828	. . . . 5 ⊢ dom (𝐵 × 𝐵) = 𝐵
25	23, 24	sseqtrdi 3967	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊‘𝐵) ⊆ 𝐵)
26	20, 25	sstrid 3928	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶 ⊆ 𝐵)
27	12	simprd 495	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹‘𝐵) ∈ 𝐵)
28	15	simprd 495	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦))
29	28	simpld 494	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊‘𝐵) We 𝐵)
30		weso 5571	. . . . . 6 ⊢ ((𝑊‘𝐵) We 𝐵 → (𝑊‘𝐵) Or 𝐵)
31	29, 30	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊‘𝐵) Or 𝐵)
32		sonr 5517	. . . . 5 ⊢ (((𝑊‘𝐵) Or 𝐵 ∧ (𝐹‘𝐵) ∈ 𝐵) → ¬ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
33	31, 27, 32	syl2anc 583	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
34	18	eleq2i 2830	. . . . 5 ⊢ ((𝐹‘𝐵) ∈ 𝐶 ↔ (𝐹‘𝐵) ∈ (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}))
35		fvex 6769	. . . . . 6 ⊢ (𝐹‘𝐵) ∈ V
36	35	eliniseg 5991	. . . . . 6 ⊢ ((𝐹‘𝐵) ∈ V → ((𝐹‘𝐵) ∈ (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}) ↔ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵)))
37	35, 36	ax-mp 5	. . . . 5 ⊢ ((𝐹‘𝐵) ∈ (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}) ↔ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
38	34, 37	bitri 274	. . . 4 ⊢ ((𝐹‘𝐵) ∈ 𝐶 ↔ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
39	33, 38	sylnibr 328	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹‘𝐵) ∈ 𝐶)
40	26, 27, 39	ssnelpssd 4043	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶 ⊊ 𝐵)
41		sneq 4568	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝐵) → {𝑦} = {(𝐹‘𝐵)})
42	41	imaeq2d 5958	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐵) → (◡(𝑊‘𝐵) “ {𝑦}) = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}))
43	42, 18	eqtr4di 2797	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐵) → (◡(𝑊‘𝐵) “ {𝑦}) = 𝐶)
44	43	fveq2d 6760	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) → (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = (𝐹‘𝐶))
45		id 22	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) → 𝑦 = (𝐹‘𝐵))
46	44, 45	eqeq12d 2754	. . . 4 ⊢ (𝑦 = (𝐹‘𝐵) → ((𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦 ↔ (𝐹‘𝐶) = (𝐹‘𝐵)))
47	28	simprd 495	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ∀𝑦 ∈ 𝐵 (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦)
48	46, 47, 27	rspcdva 3554	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹‘𝐶) = (𝐹‘𝐵))
49	48	eqcomd 2744	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹‘𝐵) = (𝐹‘𝐶))
50	17, 40, 49	3jca 1126	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883   ⊊ wpss 3884  𝒫 cpw 4530  {csn 4558  ∪ cuni 4836   class class class wbr 5070  {copab 5132   Or wor 5493   We wwe 5534   × cxp 5578  ^◡ccnv 5579  dom cdm 5580   “ cima 5583  ⟶wf 6414  ‘cfv 6418  cardccrd 9624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-en 8692  df-oi 9199  df-card 9628

This theorem is referenced by:  canthnumlem  10335  canthp1lem2  10340