Theorem canthnumlem 10404

Description: Lemma for canthnum 10405. (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 6670	. . . . 5 ⊢ (𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴 → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2		ssid 3943	. . . . . 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 10403	. . . . . 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 1449	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶)))
8	1, 7	sylan2 593	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶)))
9	8	simp3d 1143	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐹‘𝐵) = (𝐹‘𝐶))
10		simpr 485	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)
11	8	simp1d 1141	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ⊆ 𝐴)
12		elpw2g 5268	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
13	12	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
14	11, 13	mpbird 256	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ∈ 𝒫 𝐴)
15		eqid 2738	. . . . . . . . . . . 12 ⊢ 𝐵 = 𝐵
16		eqid 2738	. . . . . . . . . . . 12 ⊢ (𝑊‘𝐵) = (𝑊‘𝐵)
17	15, 16	pm3.2i 471	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))
18		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐴 ∈ 𝑉)
19	10, 1	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
20	19	ffvelrnda 6961	. . . . . . . . . . . 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 10402	. . . . . . . . . . 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 257	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐵𝑊(𝑊‘𝐵) ∧ (𝐹‘𝐵) ∈ 𝐵))
23	22	simpld 495	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵𝑊(𝑊‘𝐵))
24	3, 18	fpwwelem 10401	. . . . . . . . 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 231	. . . . . . . 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 769	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝑊‘𝐵) We 𝐵)
27		fvex 6787	. . . . . . . 8 ⊢ (𝑊‘𝐵) ∈ V
28		weeq1 5577	. . . . . . . 8 ⊢ (𝑟 = (𝑊‘𝐵) → (𝑟 We 𝐵 ↔ (𝑊‘𝐵) We 𝐵))
29	27, 28	spcev 3545	. . . . . . 7 ⊢ ((𝑊‘𝐵) We 𝐵 → ∃𝑟 𝑟 We 𝐵)
30	26, 29	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ∃𝑟 𝑟 We 𝐵)
31		ween 9791	. . . . . 6 ⊢ (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵)
32	30, 31	sylibr 233	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ∈ dom card)
33	14, 32	elind 4128	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ∈ (𝒫 𝐴 ∩ dom card))
34	8	simp2d 1142	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊊ 𝐵)
35	34	pssssd 4032	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊆ 𝐵)
36	35, 11	sstrd 3931	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊆ 𝐴)
37		elpw2g 5268	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐶 ∈ 𝒫 𝐴 ↔ 𝐶 ⊆ 𝐴))
38	37	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐶 ∈ 𝒫 𝐴 ↔ 𝐶 ⊆ 𝐴))
39	36, 38	mpbird 256	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ∈ 𝒫 𝐴)
40		ssnum 9795	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ dom card)
41	32, 35, 40	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ∈ dom card)
42	39, 41	elind 4128	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ∈ (𝒫 𝐴 ∩ dom card))
43		f1fveq 7135	. . . 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 834	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ((𝐹‘𝐵) = (𝐹‘𝐶) ↔ 𝐵 = 𝐶))
45	9, 44	mpbid 231	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 = 𝐶)
46	34	pssned 4033	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ≠ 𝐵)
47	46	necomd 2999	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ≠ 𝐶)
48	47	neneqd 2948	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ¬ 𝐵 = 𝐶)
49	45, 48	pm2.65da 814	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∀wral 3064   ∩ cin 3886   ⊆ wss 3887   ⊊ wpss 3888  𝒫 cpw 4533  {csn 4561  ∪ cuni 4839   class class class wbr 5074  {copab 5136   We wwe 5543   × cxp 5587  ^◡ccnv 5588  dom cdm 5589   “ cima 5592  ⟶wf 6429  –1-1→wf1 6430  ‘cfv 6433  cardccrd 9693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-er 8498  df-en 8734  df-dom 8735  df-oi 9269  df-card 9697

This theorem is referenced by:  canthnum  10405