Theorem canthnumlem 10621

Description: Lemma for canthnum 10622. (Contributed by Mario Carneiro, 19-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
canthnumlem	⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)

Distinct variable groups:   𝑥,𝑟,𝑦,𝐴   𝐵,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 canthnumlem

Step	Hyp	Ref	Expression
1		f1f 6764	. . . . 5 ⊢ (𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴 → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2		ssid 3961	. . . . . 6 ⊢ (𝒫 𝐴 ∩ dom card) ⊆ (𝒫 𝐴 ∩ dom card)
3		canth4.1	. . . . . . 7 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}
4		canth4.2	. . . . . . 7 ⊢ 𝐵 = ∪ dom 𝑊
5		canth4.3	. . . . . . 7 ⊢ 𝐶 = (◡(𝑊‘𝐵) “ {(𝐹‘𝐵)})
6	3, 4, 5	canth4 10620	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴 ∧ (𝒫 𝐴 ∩ dom card) ⊆ (𝒫 𝐴 ∩ dom card)) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶)))
7	2, 6	mp3an3 1474	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶)))
8	1, 7	sylan2 604	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶)))
9	8	simp3d 1160	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐹‘𝐵) = (𝐹‘𝐶))
10		simpr 489	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)
11	8	simp1d 1158	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ⊆ 𝐴)
12		elpw2g 5294	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
13	12	adantr 485	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
14	11, 13	mpbird 260	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ∈ 𝒫 𝐴)
15		eqid 2765	. . . . . . . . . . . 12 ⊢ 𝐵 = 𝐵
16		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑊‘𝐵) = (𝑊‘𝐵)
17	15, 16	pm3.2i 475	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))
18		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐴 ∈ 𝑉)
19	10, 1	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
20	19	ffvelcdmda 7069	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹‘𝑥) ∈ 𝐴)
21	3, 18, 20, 4	fpwwe 10619	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ((𝐵𝑊(𝑊‘𝐵) ∧ (𝐹‘𝐵) ∈ 𝐵) ↔ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))))
22	17, 21	mpbiri 261	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐵𝑊(𝑊‘𝐵) ∧ (𝐹‘𝐵) ∈ 𝐵))
23	22	simpld 499	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵𝑊(𝑊‘𝐵))
24	3, 18	fpwwelem 10618	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐵𝑊(𝑊‘𝐵) ↔ ((𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦))))
25	23, 24	mpbid 235	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ((𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦)))
26	25	simprld 783	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝑊‘𝐵) We 𝐵)
27		fvex 6884	. . . . . . . 8 ⊢ (𝑊‘𝐵) ∈ V
28		weeq1 5639	. . . . . . . 8 ⊢ (𝑟 = (𝑊‘𝐵) → (𝑟 We 𝐵 ↔ (𝑊‘𝐵) We 𝐵))
29	27, 28	spcev 3568	. . . . . . 7 ⊢ ((𝑊‘𝐵) We 𝐵 → ∃𝑟 𝑟 We 𝐵)
30	26, 29	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ∃𝑟 𝑟 We 𝐵)
31		ween 10007	. . . . . 6 ⊢ (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵)
32	30, 31	sylibr 237	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ∈ dom card)
33	14, 32	elind 4155	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ∈ (𝒫 𝐴 ∩ dom card))
34	8	simp2d 1159	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊊ 𝐵)
35	34	pssssd 4056	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊆ 𝐵)
36	35, 11	sstrd 3949	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊆ 𝐴)
37		elpw2g 5294	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐶 ∈ 𝒫 𝐴 ↔ 𝐶 ⊆ 𝐴))
38	37	adantr 485	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐶 ∈ 𝒫 𝐴 ↔ 𝐶 ⊆ 𝐴))
39	36, 38	mpbird 260	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ∈ 𝒫 𝐴)
40		ssnum 10011	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ dom card)
41	32, 35, 40	syl2anc 595	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ∈ dom card)
42	39, 41	elind 4155	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ∈ (𝒫 𝐴 ∩ dom card))
43		f1fveq 7250	. . . 4 ⊢ ((𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴 ∧ (𝐵 ∈ (𝒫 𝐴 ∩ dom card) ∧ 𝐶 ∈ (𝒫 𝐴 ∩ dom card))) → ((𝐹‘𝐵) = (𝐹‘𝐶) ↔ 𝐵 = 𝐶))
44	10, 33, 42, 43	syl12anc 849	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ((𝐹‘𝐵) = (𝐹‘𝐶) ↔ 𝐵 = 𝐶))
45	9, 44	mpbid 235	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 = 𝐶)
46	34	pssned 4057	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ≠ 𝐵)
47	46	necomd 3015	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ≠ 𝐶)
48	47	neneqd 2965	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ¬ 𝐵 = 𝐶)
49	45, 48	pm2.65da 828	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079   ∩ cin 3906   ⊆ wss 3907   ⊊ wpss 3908  𝒫 cpw 4558  {csn 4585  ∪ cuni 4868   class class class wbr 5105  {copab 5167   We wwe 5604   × cxp 5650  ^◡ccnv 5651  dom cdm 5652   “ cima 5655  ⟶wf 6521  –1-1→wf1 6522  ‘cfv 6525  cardccrd 9909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-er 8682  df-en 8932  df-dom 8933  df-oi 9460  df-card 9913

This theorem is referenced by:  canthnum  10622