Theorem canth4 10560

Description: An "effective" form of Cantor's theorem canth 7307. 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 2729	. . . . . . . 8 ⊢ 𝐵 = 𝐵
2		eqid 2729	. . . . . . . 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 1136	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐴 ∈ 𝑉)
6		simpl2 1193	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝐹:𝐷⟶𝐴)
7		simp3 1138	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝒫 𝐴 ∩ dom card) ⊆ 𝐷)
8	7	sselda 3937	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → 𝑥 ∈ 𝐷)
9	6, 8	ffvelcdmd 7023	. . . . . . . 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 10559	. . . . . . 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 258	. . . . . 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 10558	. . . . 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 232	. . . 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 6037	. . . . 5 ⊢ (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}) ⊆ dom (𝑊‘𝐵)
20	18, 19	eqsstri 3984	. . . 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 5849	. . . . . 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 5876	. . . . 5 ⊢ dom (𝐵 × 𝐵) = 𝐵
25	23, 24	sseqtrdi 3978	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → dom (𝑊‘𝐵) ⊆ 𝐵)
26	20, 25	sstrid 3949	. . 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 5614	. . . . . 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 5555	. . . . 5 ⊢ (((𝑊‘𝐵) Or 𝐵 ∧ (𝐹‘𝐵) ∈ 𝐵) → ¬ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
33	31, 27, 32	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
34	18	eleq2i 2820	. . . . 5 ⊢ ((𝐹‘𝐵) ∈ 𝐶 ↔ (𝐹‘𝐵) ∈ (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}))
35		fvex 6839	. . . . . 6 ⊢ (𝐹‘𝐵) ∈ V
36	35	eliniseg 6049	. . . . . 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 275	. . . 4 ⊢ ((𝐹‘𝐵) ∈ 𝐶 ↔ (𝐹‘𝐵)(𝑊‘𝐵)(𝐹‘𝐵))
39	33, 38	sylnibr 329	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → ¬ (𝐹‘𝐵) ∈ 𝐶)
40	26, 27, 39	ssnelpssd 4068	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → 𝐶 ⊊ 𝐵)
41		sneq 4589	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝐵) → {𝑦} = {(𝐹‘𝐵)})
42	41	imaeq2d 6015	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐵) → (◡(𝑊‘𝐵) “ {𝑦}) = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)}))
43	42, 18	eqtr4di 2782	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐵) → (◡(𝑊‘𝐵) “ {𝑦}) = 𝐶)
44	43	fveq2d 6830	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) → (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = (𝐹‘𝐶))
45		id 22	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐵) → 𝑦 = (𝐹‘𝐵))
46	44, 45	eqeq12d 2745	. . . 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 3580	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹‘𝐶) = (𝐹‘𝐵))
49	48	eqcomd 2735	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐷⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ 𝐷) → (𝐹‘𝐵) = (𝐹‘𝐶))
50	17, 40, 49	3jca 1128	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3438   ∩ cin 3904   ⊆ wss 3905   ⊊ wpss 3906  𝒫 cpw 4553  {csn 4579  ∪ cuni 4861   class class class wbr 5095  {copab 5157   Or wor 5530   We wwe 5575   × cxp 5621  ^◡ccnv 5622  dom cdm 5623   “ cima 5626  ⟶wf 6482  ‘cfv 6486  cardccrd 9850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7310  df-ov 7356  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-en 8880  df-oi 9421  df-card 9854

This theorem is referenced by:  canthnumlem  10561  canthp1lem2  10566