Theorem fin4en1 10315

Description: Dedekind finite is a cardinal property. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
fin4en1	⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinIV → 𝐵 ∈ FinIV))

Proof of Theorem fin4en1

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ensym 9013	. 2 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
2		bren 8966	. . . 4 ⊢ (𝐵 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐴)
3		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → 𝑥 ⊊ 𝐵)
4		f1of1 6820	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵–1-1→𝐴)
5		pssss 4049	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊊ 𝐵 → 𝑥 ⊆ 𝐵)
6		ssid 3956	. . . . . . . . . . . . . 14 ⊢ 𝐵 ⊆ 𝐵
7	5, 6	jctir 530	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊊ 𝐵 → (𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵))
8		f1imapss 7267	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ (𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵)) → ((𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵) ↔ 𝑥 ⊊ 𝐵))
9	4, 7, 8	syl2an 608	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → ((𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵) ↔ 𝑥 ⊊ 𝐵))
10	3, 9	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵))
11		imadmrn 6070	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
12		f1odm 6825	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → dom 𝑓 = 𝐵)
13	12	imaeq2d 6060	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ 𝐵))
14		dff1o5 6831	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 ↔ (𝑓:𝐵–1-1→𝐴 ∧ ran 𝑓 = 𝐴))
15	14	simprbi 503	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → ran 𝑓 = 𝐴)
16	11, 13, 15	3eqtr3a 2821	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝑓 “ 𝐵) = 𝐴)
17	16	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝐵) = 𝐴)
18	17	psseq2d 4047	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → ((𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵) ↔ (𝑓 “ 𝑥) ⊊ 𝐴))
19	10, 18	mpbid 235	. . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝑥) ⊊ 𝐴)
20	19	adantrr 730	. . . . . . . . 9 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ⊊ 𝐴)
21		vex 3457	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
22	21	f1imaen 9027	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝑥) ≈ 𝑥)
23	4, 5, 22	syl2an 608	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝑥) ≈ 𝑥)
24	23	adantrr 730	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝑥)
25		simprr 785	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝑥 ≈ 𝐵)
26		entr 9016	. . . . . . . . . . 11 ⊢ (((𝑓 “ 𝑥) ≈ 𝑥 ∧ 𝑥 ≈ 𝐵) → (𝑓 “ 𝑥) ≈ 𝐵)
27	24, 25, 26	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝐵)
28		vex 3457	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
29		f1oen3g 8976	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐵–1-1-onto→𝐴) → 𝐵 ≈ 𝐴)
30	28, 29	mpan 703	. . . . . . . . . . 11 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝐵 ≈ 𝐴)
31	30	adantr 486	. . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝐵 ≈ 𝐴)
32		entr 9016	. . . . . . . . . 10 ⊢ (((𝑓 “ 𝑥) ≈ 𝐵 ∧ 𝐵 ≈ 𝐴) → (𝑓 “ 𝑥) ≈ 𝐴)
33	27, 31, 32	syl2anc 596	. . . . . . . . 9 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝐴)
34		fin4i 10304	. . . . . . . . 9 ⊢ (((𝑓 “ 𝑥) ⊊ 𝐴 ∧ (𝑓 “ 𝑥) ≈ 𝐴) → ¬ 𝐴 ∈ FinIV)
35	20, 33, 34	syl2anc 596	. . . . . . . 8 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → ¬ 𝐴 ∈ FinIV)
36	35	ex 418	. . . . . . 7 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → ((𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵) → ¬ 𝐴 ∈ FinIV))
37	36	exlimdv 1966	. . . . . 6 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵) → ¬ 𝐴 ∈ FinIV))
38	37	con2d 135	. . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
39	38	exlimiv 1963	. . . 4 ⊢ (∃𝑓 𝑓:𝐵–1-1-onto→𝐴 → (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
40	2, 39	sylbi 220	. . 3 ⊢ (𝐵 ≈ 𝐴 → (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
41		relen 8961	. . . . 5 ⊢ Rel ≈
42	41	brrelex1i 5715	. . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ∈ V)
43		isfin4 10303	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
44	42, 43	syl 18	. . 3 ⊢ (𝐵 ≈ 𝐴 → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
45	40, 44	sylibrd 262	. 2 ⊢ (𝐵 ≈ 𝐴 → (𝐴 ∈ FinIV → 𝐵 ∈ FinIV))
46	1, 45	syl 18	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinIV → 𝐵 ∈ FinIV))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902   ⊊ wpss 3903   class class class wbr 5107  dom cdm 5659  ran crn 5660   “ cima 5662  –1-1→wf1 6534  –1-1-onto→wf1o 6536   ≈ cen 8953  Fin^IVcfin4 10286

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-er 8700  df-en 8957  df-fin4 10293

This theorem is used by:  domfin4  10317  isfin4p1  10321