Theorem canthnumlem 10763

Description: Lemma for canthnum 10764. (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)⟶1-1𝐴)

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 6803	. . . . 5 ⊢ (𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴 → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2		ssid 3935	. . . . . 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 10762	. . . . . 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 1480	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶)))
8	1, 7	sylan2 605	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → (𝐵 ⊆ 𝐴 ∧ 𝐶 ⊊ 𝐵 ∧ (𝐹‘𝐵) = (𝐹‘𝐶)))
9	8	simp3d 1162	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → (𝐹‘𝐵) = (𝐹‘𝐶))
10		simpr 490	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴)
11	8	simp1d 1160	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐵 ⊆ 𝐴)
12		elpw2g 5300	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
13	12	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
14	11, 13	mpbird 260	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐵 ∈ 𝒫 𝐴)
15		eqid 2763	. . . . . . . . . . . 12 ⊢ 𝐵 = 𝐵
16		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑊‘𝐵) = (𝑊‘𝐵)
17	15, 16	pm3.2i 476	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))
18		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐴 ∈ 𝑉)
19	10, 1	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
20	19	ffvelcdmda 7111	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) ∧ 𝑥 ∈ (𝒫 𝐴 ∩ dom card)) → (𝐹‘𝑥) ∈ 𝐴)
21	3, 18, 20, 4	fpwwe 10761	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → ((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)⟶1-1𝐴) → (𝐵𝑊(𝑊‘𝐵) ∧ (𝐹‘𝐵) ∈ 𝐵))
23	22	simpld 500	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐵𝑊(𝑊‘𝐵))
24	3, 18	fpwwelem 10760	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → (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)⟶1-1𝐴) → ((𝐵 ⊆ 𝐴 ∧ (𝑊‘𝐵) ⊆ (𝐵 × 𝐵)) ∧ ((𝑊‘𝐵) We 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐹‘(◡(𝑊‘𝐵) “ {𝑦})) = 𝑦)))
26	25	simprld 784	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → (𝑊‘𝐵) We 𝐵)
27		fvex 6924	. . . . . . . 8 ⊢ (𝑊‘𝐵) ∈ V
28		weeq1 5643	. . . . . . . 8 ⊢ (𝑟 = (𝑊‘𝐵) → (𝑟 We 𝐵 ↔ (𝑊‘𝐵) We 𝐵))
29	27, 28	spcev 3564	. . . . . . 7 ⊢ ((𝑊‘𝐵) We 𝐵 → ∃𝑟 𝑟 We 𝐵)
30	26, 29	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → ∃𝑟 𝑟 We 𝐵)
31		ween 10144	. . . . . 6 ⊢ (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵)
32	30, 31	sylibr 237	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐵 ∈ dom card)
33	14, 32	elind 4100	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐵 ∈ (𝒫 𝐴 ∩ dom card))
34	8	simp2d 1161	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐶 ⊊ 𝐵)
35	34	pssssd 4030	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐶 ⊆ 𝐵)
36	35, 11	sstrd 3923	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐶 ⊆ 𝐴)
37		elpw2g 5300	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐶 ∈ 𝒫 𝐴 ↔ 𝐶 ⊆ 𝐴))
38	37	adantr 486	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → (𝐶 ∈ 𝒫 𝐴 ↔ 𝐶 ⊆ 𝐴))
39	36, 38	mpbird 260	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐶 ∈ 𝒫 𝐴)
40		ssnum 10148	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ dom card)
41	32, 35, 40	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐶 ∈ dom card)
42	39, 41	elind 4100	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐶 ∈ (𝒫 𝐴 ∩ dom card))
43		f1fveq 7292	. . . 4 ⊢ ((𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴 ∧ (𝐵 ∈ (𝒫 𝐴 ∩ dom card) ∧ 𝐶 ∈ (𝒫 𝐴 ∩ dom card))) → ((𝐹‘𝐵) = (𝐹‘𝐶) ↔ 𝐵 = 𝐶))
44	10, 33, 42, 43	syl12anc 850	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → ((𝐹‘𝐵) = (𝐹‘𝐶) ↔ 𝐵 = 𝐶))
45	9, 44	mpbid 235	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐵 = 𝐶)
46	34	pssned 4031	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐶 ≠ 𝐵)
47	46	necomd 3014	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → 𝐵 ≠ 𝐶)
48	47	neneqd 2964	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴) → ¬ 𝐵 = 𝐶)
49	45, 48	pm2.65da 829	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)⟶1-1𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ⊊ wpss 4016   ∩ cin 4092  𝒫 cpw 4560  {csn 4588  ∪ cuni 4870   class class class wbr 5106  {copab 5171   We wwe 5608   × cxp 5654  ^◡ccnv 5655  dom cdm 5656   “ cima 5659  ⟶wf 6557  ⟶1-1wf1 6558  ‘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-er 8748  df-en 9005  df-dom 9006  df-oi 9534  df-card 10050

This theorem is used by:  canthnum  10764