Theorem fin4en1 10238

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 8951	. 2 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
2		bren 8905	. . . 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 6781	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵–1-1→𝐴)
5		pssss 4057	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊊ 𝐵 → 𝑥 ⊆ 𝐵)
6		ssid 3966	. . . . . . . . . . . . . 14 ⊢ 𝐵 ⊆ 𝐵
7	5, 6	jctir 520	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊊ 𝐵 → (𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵))
8		f1imapss 7223	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ (𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵)) → ((𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵) ↔ 𝑥 ⊊ 𝐵))
9	4, 7, 8	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → ((𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵) ↔ 𝑥 ⊊ 𝐵))
10	3, 9	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵))
11		imadmrn 6030	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
12		f1odm 6786	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → dom 𝑓 = 𝐵)
13	12	imaeq2d 6020	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ 𝐵))
14		dff1o5 6791	. . . . . . . . . . . . . . 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 2788	. . . . . . . . . . . . 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 4055	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → ((𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵) ↔ (𝑓 “ 𝑥) ⊊ 𝐴))
19	10, 18	mpbid 232	. . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝑥) ⊊ 𝐴)
20	19	adantrr 717	. . . . . . . . 9 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ⊊ 𝐴)
21		vex 3448	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
22	21	f1imaen 8965	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝑥) ≈ 𝑥)
23	4, 5, 22	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝑥) ≈ 𝑥)
24	23	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝑥)
25		simprr 772	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝑥 ≈ 𝐵)
26		entr 8954	. . . . . . . . . . 11 ⊢ (((𝑓 “ 𝑥) ≈ 𝑥 ∧ 𝑥 ≈ 𝐵) → (𝑓 “ 𝑥) ≈ 𝐵)
27	24, 25, 26	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝐵)
28		vex 3448	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
29		f1oen3g 8915	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐵–1-1-onto→𝐴) → 𝐵 ≈ 𝐴)
30	28, 29	mpan 690	. . . . . . . . . . 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 8954	. . . . . . . . . 10 ⊢ (((𝑓 “ 𝑥) ≈ 𝐵 ∧ 𝐵 ≈ 𝐴) → (𝑓 “ 𝑥) ≈ 𝐴)
33	27, 31, 32	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝐴)
34		fin4i 10227	. . . . . . . . 9 ⊢ (((𝑓 “ 𝑥) ⊊ 𝐴 ∧ (𝑓 “ 𝑥) ≈ 𝐴) → ¬ 𝐴 ∈ FinIV)
35	20, 33, 34	syl2anc 584	. . . . . . . 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 1933	. . . . . 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 1930	. . . 4 ⊢ (∃𝑓 𝑓:𝐵–1-1-onto→𝐴 → (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
40	2, 39	sylbi 217	. . 3 ⊢ (𝐵 ≈ 𝐴 → (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
41		relen 8900	. . . . 5 ⊢ Rel ≈
42	41	brrelex1i 5687	. . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ∈ V)
43		isfin4 10226	. . . 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 259	. 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 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3444   ⊆ wss 3911   ⊊ wpss 3912   class class class wbr 5102  dom cdm 5631  ran crn 5632   “ cima 5634  –1-1→wf1 6496  –1-1-onto→wf1o 6498   ≈ cen 8892  Fin^IVcfin4 10209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-er 8648  df-en 8896  df-fin4 10216

This theorem is referenced by:  domfin4  10240  isfin4p1  10244