Theorem canth4 10762

Description: An "effective" form of Cantor's theorem canth 7400. 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 2763	. . . . . . . 8 ⊢ 𝐵 = 𝐵
2		eqid 2763	. . . . . . . 8 ⊢ (𝑊‘𝐵) = (𝑊‘𝐵)
3	1, 2	pm3.2i 476	. . . . . . 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 1154	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐴 ∈ 𝑉)
6		simpl2 1211	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝐹:𝐷⟶𝐴)
7		simp3 1156	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝒫 𝐴 ∩ dom card) ⊆ 𝐷)
8	7	sselda 3913	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝑥 ∈ 𝐷)
9	6, 8	ffvelcdmd 7112	. . . . . . . 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 10761	. . . . . . 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 261	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵𝑊(𝑊‘𝐵) ∧ (𝐹‘𝐵) ∈ 𝐵))
13	12	simpld 500	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵𝑊(𝑊‘𝐵))
14	4, 5	fpwwelem 10760	. . . . 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 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)))
16	15	simpld 500	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)))
17	16	simpld 500	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐵 ⊆ 𝐴)
18		canth4.3	. . . . 5 ⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)})
19		cnvimass 6212	. . . . 5 ⊢ (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}) ⊆ dom (𝑊‘𝐵)
20	18, 19	eqsstri 3959	. . . 4 ⊢ 𝐶 ⊆ dom (𝑊‘𝐵)
21	16	simprd 501	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊‘𝐵) ⊆ (𝐵 × 𝐵))
22		dmss 5892	. . . . . 6 ⊢ ((𝑊‘𝐵) ⊆ (𝐵 × 𝐵) → dom (𝑊‘𝐵) ⊆ dom (𝐵 × 𝐵))
23	21, 22	syl 18	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊‘𝐵) ⊆ dom (𝐵 × 𝐵))
24		dmxpid 5920	. . . . 5 ⊢ dom (𝐵 × 𝐵) = 𝐵
25	23, 24	sseqtrdi 3953	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊‘𝐵) ⊆ 𝐵)
26	20, 25	sstrid 3924	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶 ⊆ 𝐵)
27	12	simprd 501	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹‘𝐵) ∈ 𝐵)
28	15	simprd 501	. . . . . . 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 500	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊‘𝐵) We 𝐵)
30		weso 5647	. . . . . 6 ⊢ ((𝑊‘𝐵) We 𝐵 → (𝑊‘𝐵) Or 𝐵)
31	29, 30	syl 18	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝑊‘𝐵) Or 𝐵)
32		sonr 5588	. . . . 5 ⊢ (((𝑊‘𝐵) Or 𝐵 ∧ (𝐹‘𝐵) ∈ 𝐵) → ¬ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
33	31, 27, 32	syl2anc 596	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
34	18	eleq2i 2856	. . . . 5 ⊢ ((𝐹‘𝐵) ∈ 𝐶 ↔ (𝐹‘𝐵) ∈ (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}))
35		fvex 6924	. . . . . 6 ⊢ (𝐹‘𝐵) ∈ V
36	35	eliniseg 6105	. . . . . 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 278	. . . 4 ⊢ ((𝐹‘𝐵) ∈ 𝐶 ↔ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
39	33, 38	sylnibr 332	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹‘𝐵) ∈ 𝐶)
40	26, 27, 39	ssnelpssd 4046	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶 ⊊ 𝐵)
41		sneq 4594	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝐵) → {𝑦} = {(𝐹‘𝐵)})
42	41	imaeq2d 6065	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐵) → (◡(𝑊‘𝐵) “ {𝑦}) = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}))
43	42, 18	eqtr4di 2816	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐵) → (◡(𝑊‘𝐵) “ {𝑦}) = 𝐶)
44	43	fveq2d 6915	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) → (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = (𝐹‘𝐶))
45		id 23	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) → 𝑦 = (𝐹‘𝐵))
46	44, 45	eqeq12d 2779	. . . 4 ⊢ (𝑦 = (𝐹‘𝐵) → ((𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦 ↔ (𝐹‘𝐶) = (𝐹‘𝐵)))
47	28	simprd 501	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ∀𝑦 ∈ 𝐵 (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦)
48	46, 47, 27	rspcdva 3581	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹‘𝐶) = (𝐹‘𝐵))
49	48	eqcomd 2769	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹‘𝐵) = (𝐹‘𝐶))
50	17, 40, 49	3jca 1146	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ⊊ wpss 4016   ∩ cin 4092  𝒫 cpw 4560  {csn 4588  ∪ cuni 4870   class class class wbr 5106  {copab 5171   Or wor 5563   We wwe 5608   × cxp 5654  ^◡ccnv 5655  dom cdm 5656   “ cima 5659  ⟶wf 6557  ‘cfv 6561  cardccrd 10046

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7403  df-ov 7449  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-en 9005  df-oi 9534  df-card 10050

This theorem is used by:  canthnumlem  10763  canthp1lem2  10768