Theorem canth4 10058

Description: An "effective" form of Cantor's theorem canth 7090. 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 2798	. . . . . . . 8 ⊢ 𝐵 = 𝐵
2		eqid 2798	. . . . . . . 8 ⊢ (𝑊‘𝐵) = (𝑊‘𝐵)
3	1, 2	pm3.2i 474	. . . . . . 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		elex 3459	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
6	5	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐴 ∈ V)
7		simpl2 1189	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝐹:𝐷⟶𝐴)
8		simp3 1135	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝒫 𝐴 ∩ dom card) ⊆ 𝐷)
9	8	sselda 3915	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝑥 ∈ 𝐷)
10	7, 9	ffvelrnd 6829	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹‘𝑥) ∈ 𝐴)
11		canth4.2	. . . . . . . 8 ⊢ 𝐵 = ∪ dom 𝑊
12	4, 6, 10, 11	fpwwe 10057	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝐵𝑊(𝑊‘𝐵) ∧ (𝐹‘𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))))
13	3, 12	mpbiri 261	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝑊(𝑊‘𝐵) ∧ (𝐹‘𝐵) ∈ 𝐵))
14	13	simpld 498	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵𝑊(𝑊‘𝐵))
15	4, 6	fpwwelem 10056	. . . . 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))))
16	14, 15	mpbid 235	. . . 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)))
17	16	simpld 498	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)))
18	17	simpld 498	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵 ⊆ 𝐴)
19		canth4.3	. . . . 5 ⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)})
20		cnvimass 5916	. . . . 5 ⊢ (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}) ⊆ dom (𝑊‘𝐵)
21	19, 20	eqsstri 3949	. . . 4 ⊢ 𝐶 ⊆ dom (𝑊‘𝐵)
22	17	simprd 499	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊‘𝐵) ⊆ (𝐵 × 𝐵))
23		dmss 5735	. . . . . 6 ⊢ ((𝑊‘𝐵) ⊆ (𝐵 × 𝐵) → dom (𝑊‘𝐵) ⊆ dom (𝐵 × 𝐵))
24	22, 23	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊‘𝐵) ⊆ dom (𝐵 × 𝐵))
25		dmxpid 5764	. . . . 5 ⊢ dom (𝐵 × 𝐵) = 𝐵
26	24, 25	sseqtrdi 3965	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊‘𝐵) ⊆ 𝐵)
27	21, 26	sstrid 3926	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶 ⊆ 𝐵)
28	13	simprd 499	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹‘𝐵) ∈ 𝐵)
29	16	simprd 499	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦))
30	29	simpld 498	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊‘𝐵) We 𝐵)
31		weso 5510	. . . . . 6 ⊢ ((𝑊‘𝐵) We 𝐵 → (𝑊‘𝐵) Or 𝐵)
32	30, 31	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊‘𝐵) Or 𝐵)
33		sonr 5460	. . . . 5 ⊢ (((𝑊‘𝐵) Or 𝐵 ∧ (𝐹‘𝐵) ∈ 𝐵) → ¬ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
34	32, 28, 33	syl2anc 587	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
35	19	eleq2i 2881	. . . . 5 ⊢ ((𝐹‘𝐵) ∈ 𝐶 ↔ (𝐹‘𝐵) ∈ (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}))
36		fvex 6658	. . . . . 6 ⊢ (𝐹‘𝐵) ∈ V
37	36	eliniseg 5925	. . . . . 6 ⊢ ((𝐹‘𝐵) ∈ V → ((𝐹‘𝐵) ∈ (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}) ↔ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵)))
38	36, 37	ax-mp 5	. . . . 5 ⊢ ((𝐹‘𝐵) ∈ (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}) ↔ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
39	35, 38	bitri 278	. . . 4 ⊢ ((𝐹‘𝐵) ∈ 𝐶 ↔ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
40	34, 39	sylnibr 332	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹‘𝐵) ∈ 𝐶)
41	27, 28, 40	ssnelpssd 4040	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶 ⊊ 𝐵)
42		sneq 4535	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝐵) → {𝑦} = {(𝐹‘𝐵)})
43	42	imaeq2d 5896	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐵) → (◡(𝑊‘𝐵) “ {𝑦}) = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}))
44	43, 19	eqtr4di 2851	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐵) → (◡(𝑊‘𝐵) “ {𝑦}) = 𝐶)
45	44	fveq2d 6649	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) → (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = (𝐹‘𝐶))
46		id 22	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) → 𝑦 = (𝐹‘𝐵))
47	45, 46	eqeq12d 2814	. . . 4 ⊢ (𝑦 = (𝐹‘𝐵) → ((𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦 ↔ (𝐹‘𝐶) = (𝐹‘𝐵)))
48	29	simprd 499	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ∀𝑦 ∈ 𝐵 (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦)
49	47, 48, 28	rspcdva 3573	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹‘𝐶) = (𝐹‘𝐵))
50	49	eqcomd 2804	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹‘𝐵) = (𝐹‘𝐶))
51	18, 41, 50	3jca 1125	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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881   ⊊ wpss 3882  𝒫 cpw 4497  {csn 4525  ∪ cuni 4800   class class class wbr 5030  {copab 5092   Or wor 5437   We wwe 5477   × cxp 5517  ^◡ccnv 5518  dom cdm 5519   “ cima 5522  ⟶wf 6320  ‘cfv 6324  cardccrd 9348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-1st 7671  df-wrecs 7930  df-recs 7991  df-en 8493  df-oi 8958  df-card 9352

This theorem is referenced by:  canthnumlem  10059  canthp1lem2  10064