Theorem canthnumlem 10536

Description: Lemma for canthnum 10537. (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 6719	. . . . 5 ⊢ (𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴 → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2		ssid 3957	. . . . . 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 10535	. . . . . 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 1452	. . . . 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 1144	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐹‘𝐵) = (𝐹‘𝐶))
10		simpr 484	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)
11	8	simp1d 1142	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ⊆ 𝐴)
12		elpw2g 5271	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
13	12	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
14	11, 13	mpbird 257	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ∈ 𝒫 𝐴)
15		eqid 2731	. . . . . . . . . . . 12 ⊢ 𝐵 = 𝐵
16		eqid 2731	. . . . . . . . . . . 12 ⊢ (𝑊‘𝐵) = (𝑊‘𝐵)
17	15, 16	pm3.2i 470	. . . . . . . . . . 11 ⊢ (𝐵 = 𝐵 ∧ (𝑊‘𝐵) = (𝑊‘𝐵))
18		simpl 482	. . . . . . . . . . . 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	ffvelcdmda 7017	. . . . . . . . . . . 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 10534	. . . . . . . . . . 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 258	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐵𝑊(𝑊‘𝐵) ∧ (𝐹‘𝐵) ∈ 𝐵))
23	22	simpld 494	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵𝑊(𝑊‘𝐵))
24	3, 18	fpwwelem 10533	. . . . . . . . 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 232	. . . . . . . 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 771	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝑊‘𝐵) We 𝐵)
27		fvex 6835	. . . . . . . 8 ⊢ (𝑊‘𝐵) ∈ V
28		weeq1 5603	. . . . . . . 8 ⊢ (𝑟 = (𝑊‘𝐵) → (𝑟 We 𝐵 ↔ (𝑊‘𝐵) We 𝐵))
29	27, 28	spcev 3561	. . . . . . 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 9923	. . . . . 6 ⊢ (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵)
32	30, 31	sylibr 234	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ∈ dom card)
33	14, 32	elind 4150	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ∈ (𝒫 𝐴 ∩ dom card))
34	8	simp2d 1143	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊊ 𝐵)
35	34	pssssd 4050	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊆ 𝐵)
36	35, 11	sstrd 3945	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊆ 𝐴)
37		elpw2g 5271	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝐶 ∈ 𝒫 𝐴 ↔ 𝐶 ⊆ 𝐴))
38	37	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝐶 ∈ 𝒫 𝐴 ↔ 𝐶 ⊆ 𝐴))
39	36, 38	mpbird 257	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ∈ 𝒫 𝐴)
40		ssnum 9927	. . . . . 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 4150	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ∈ (𝒫 𝐴 ∩ dom card))
43		f1fveq 7196	. . . 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 836	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ((𝐹‘𝐵) = (𝐹‘𝐶) ↔ 𝐵 = 𝐶))
45	9, 44	mpbid 232	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 = 𝐶)
46	34	pssned 4051	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ≠ 𝐵)
47	46	necomd 2983	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ≠ 𝐶)
48	47	neneqd 2933	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ¬ 𝐵 = 𝐶)
49	45, 48	pm2.65da 816	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∀wral 3047   ∩ cin 3901   ⊆ wss 3902   ⊊ wpss 3903  𝒫 cpw 4550  {csn 4576  ∪ cuni 4859   class class class wbr 5091  {copab 5153   We wwe 5568   × cxp 5614  ^◡ccnv 5615  dom cdm 5616   “ cima 5619  ⟶wf 6477  –1-1→wf1 6478  ‘cfv 6481  cardccrd 9825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-er 8622  df-en 8870  df-dom 8871  df-oi 9396  df-card 9829

This theorem is referenced by:  canthnum  10537