Theorem fin4en1 9996

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 8744	. 2 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
2		bren 8701	. . . 4 ⊢ (𝐵 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐴)
3		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → 𝑥 ⊊ 𝐵)
4		f1of1 6699	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵–1-1→𝐴)
5		pssss 4026	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊊ 𝐵 → 𝑥 ⊆ 𝐵)
6		ssid 3939	. . . . . . . . . . . . . 14 ⊢ 𝐵 ⊆ 𝐵
7	5, 6	jctir 520	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊊ 𝐵 → (𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵))
8		f1imapss 7120	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ (𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵)) → ((𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵) ↔ 𝑥 ⊊ 𝐵))
9	4, 7, 8	syl2an 595	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → ((𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵) ↔ 𝑥 ⊊ 𝐵))
10	3, 9	mpbird 256	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵))
11		imadmrn 5968	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
12		f1odm 6704	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → dom 𝑓 = 𝐵)
13	12	imaeq2d 5958	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ 𝐵))
14		dff1o5 6709	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 ↔ (𝑓:𝐵–1-1→𝐴 ∧ ran 𝑓 = 𝐴))
15	14	simprbi 496	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → ran 𝑓 = 𝐴)
16	11, 13, 15	3eqtr3a 2803	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝑓 “ 𝐵) = 𝐴)
17	16	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝐵) = 𝐴)
18	17	psseq2d 4024	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → ((𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵) ↔ (𝑓 “ 𝑥) ⊊ 𝐴))
19	10, 18	mpbid 231	. . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝑥) ⊊ 𝐴)
20	19	adantrr 713	. . . . . . . . 9 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ⊊ 𝐴)
21		vex 3426	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
22	21	f1imaen 8757	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝑥) ≈ 𝑥)
23	4, 5, 22	syl2an 595	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝑥) ≈ 𝑥)
24	23	adantrr 713	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝑥)
25		simprr 769	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝑥 ≈ 𝐵)
26		entr 8747	. . . . . . . . . . 11 ⊢ (((𝑓 “ 𝑥) ≈ 𝑥 ∧ 𝑥 ≈ 𝐵) → (𝑓 “ 𝑥) ≈ 𝐵)
27	24, 25, 26	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝐵)
28		vex 3426	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
29		f1oen3g 8709	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐵–1-1-onto→𝐴) → 𝐵 ≈ 𝐴)
30	28, 29	mpan 686	. . . . . . . . . . 11 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝐵 ≈ 𝐴)
31	30	adantr 480	. . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝐵 ≈ 𝐴)
32		entr 8747	. . . . . . . . . 10 ⊢ (((𝑓 “ 𝑥) ≈ 𝐵 ∧ 𝐵 ≈ 𝐴) → (𝑓 “ 𝑥) ≈ 𝐴)
33	27, 31, 32	syl2anc 583	. . . . . . . . 9 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝐴)
34		fin4i 9985	. . . . . . . . 9 ⊢ (((𝑓 “ 𝑥) ⊊ 𝐴 ∧ (𝑓 “ 𝑥) ≈ 𝐴) → ¬ 𝐴 ∈ FinIV)
35	20, 33, 34	syl2anc 583	. . . . . . . 8 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → ¬ 𝐴 ∈ FinIV)
36	35	ex 412	. . . . . . 7 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → ((𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵) → ¬ 𝐴 ∈ FinIV))
37	36	exlimdv 1937	. . . . . 6 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵) → ¬ 𝐴 ∈ FinIV))
38	37	con2d 134	. . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
39	38	exlimiv 1934	. . . 4 ⊢ (∃𝑓 𝑓:𝐵–1-1-onto→𝐴 → (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
40	2, 39	sylbi 216	. . 3 ⊢ (𝐵 ≈ 𝐴 → (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
41		relen 8696	. . . . 5 ⊢ Rel ≈
42	41	brrelex1i 5634	. . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ∈ V)
43		isfin4 9984	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
44	42, 43	syl 17	. . 3 ⊢ (𝐵 ≈ 𝐴 → (𝐵 ∈ FinIV ↔ ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
45	40, 44	sylibrd 258	. 2 ⊢ (𝐵 ≈ 𝐴 → (𝐴 ∈ FinIV → 𝐵 ∈ FinIV))
46	1, 45	syl 17	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinIV → 𝐵 ∈ FinIV))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883   ⊊ wpss 3884   class class class wbr 5070  dom cdm 5580  ran crn 5581   “ cima 5583  –1-1→wf1 6415  –1-1-onto→wf1o 6417   ≈ cen 8688  Fin^IVcfin4 9967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-er 8456  df-en 8692  df-fin4 9974

This theorem is referenced by:  domfin4  9998  isfin4p1  10002