Theorem canthnumlem 10639

Description: Lemma for canthnum 10640. (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 6784	. . . . 5 ⊢ (𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴 → 𝐹:(𝒫 𝐴 ∩ dom card)⟶𝐴)
2		ssid 4003	. . . . . 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 10638	. . . . . 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 1450	. . . . 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 485	. . . 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 5343	. . . . . . 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 2732	. . . . . . . . . . . 12 ⊢ 𝐵 = 𝐵
16		eqid 2732	. . . . . . . . . . . 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	ffvelcdmda 7083	. . . . . . . . . . . 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 10637	. . . . . . . . . . 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 10636	. . . . . . . . 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 770	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → (𝑊‘𝐵) We 𝐵)
27		fvex 6901	. . . . . . . 8 ⊢ (𝑊‘𝐵) ∈ V
28		weeq1 5663	. . . . . . . 8 ⊢ (𝑟 = (𝑊‘𝐵) → (𝑟 We 𝐵 ↔ (𝑊‘𝐵) We 𝐵))
29	27, 28	spcev 3596	. . . . . . 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 10026	. . . . . 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 4193	. . . 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 4096	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊆ 𝐵)
36	35, 11	sstrd 3991	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ⊆ 𝐴)
37		elpw2g 5343	. . . . . . 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 10030	. . . . . 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 4193	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ∈ (𝒫 𝐴 ∩ dom card))
43		f1fveq 7257	. . . 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 835	. . 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 4097	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐶 ≠ 𝐵)
47	46	necomd 2996	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → 𝐵 ≠ 𝐶)
48	47	neneqd 2945	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴) → ¬ 𝐵 = 𝐶)
49	45, 48	pm2.65da 815	1 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐹:(𝒫 𝐴 ∩ dom card)–1-1→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3061   ∩ cin 3946   ⊆ wss 3947   ⊊ wpss 3948  𝒫 cpw 4601  {csn 4627  ∪ cuni 4907   class class class wbr 5147  {copab 5209   We wwe 5629   × cxp 5673  ^◡ccnv 5674  dom cdm 5675   “ cima 5678  ⟶wf 6536  –1-1→wf1 6537  ‘cfv 6540  cardccrd 9926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-er 8699  df-en 8936  df-dom 8937  df-oi 9501  df-card 9930

This theorem is referenced by:  canthnum  10640