Theorem fin4en1 10231

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 8950	. 2 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
2		bren 8903	. . . 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 6779	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵–1-1→𝐴)
5		pssss 4038	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊊ 𝐵 → 𝑥 ⊆ 𝐵)
6		ssid 3944	. . . . . . . . . . . . . 14 ⊢ 𝐵 ⊆ 𝐵
7	5, 6	jctir 520	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊊ 𝐵 → (𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵))
8		f1imapss 7221	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ (𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵)) → ((𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵) ↔ 𝑥 ⊊ 𝐵))
9	4, 7, 8	syl2an 597	. . . . . . . . . . . 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 6035	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
12		f1odm 6784	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → dom 𝑓 = 𝐵)
13	12	imaeq2d 6025	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ 𝐵))
14		dff1o5 6789	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 ↔ (𝑓:𝐵–1-1→𝐴 ∧ ran 𝑓 = 𝐴))
15	14	simprbi 497	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → ran 𝑓 = 𝐴)
16	11, 13, 15	3eqtr3a 2795	. . . . . . . . . . . . 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 4036	. . . . . . . . . . 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 718	. . . . . . . . 9 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ⊊ 𝐴)
21		vex 3433	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
22	21	f1imaen 8964	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝑥) ≈ 𝑥)
23	4, 5, 22	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝑥) ≈ 𝑥)
24	23	adantrr 718	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝑥)
25		simprr 773	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝑥 ≈ 𝐵)
26		entr 8953	. . . . . . . . . . 11 ⊢ (((𝑓 “ 𝑥) ≈ 𝑥 ∧ 𝑥 ≈ 𝐵) → (𝑓 “ 𝑥) ≈ 𝐵)
27	24, 25, 26	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝐵)
28		vex 3433	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
29		f1oen3g 8913	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐵–1-1-onto→𝐴) → 𝐵 ≈ 𝐴)
30	28, 29	mpan 691	. . . . . . . . . . 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 8953	. . . . . . . . . 10 ⊢ (((𝑓 “ 𝑥) ≈ 𝐵 ∧ 𝐵 ≈ 𝐴) → (𝑓 “ 𝑥) ≈ 𝐴)
33	27, 31, 32	syl2anc 585	. . . . . . . . 9 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝐴)
34		fin4i 10220	. . . . . . . . 9 ⊢ (((𝑓 “ 𝑥) ⊊ 𝐴 ∧ (𝑓 “ 𝑥) ≈ 𝐴) → ¬ 𝐴 ∈ FinIV)
35	20, 33, 34	syl2anc 585	. . . . . . . 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 1935	. . . . . 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 1932	. . . 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 8898	. . . . 5 ⊢ Rel ≈
42	41	brrelex1i 5687	. . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ∈ V)
43		isfin4 10219	. . . 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 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3429   ⊆ wss 3889   ⊊ wpss 3890   class class class wbr 5085  dom cdm 5631  ran crn 5632   “ cima 5634  –1-1→wf1 6495  –1-1-onto→wf1o 6497   ≈ cen 8890  Fin^IVcfin4 10202

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  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 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-er 8643  df-en 8894  df-fin4 10209

This theorem is referenced by:  domfin4  10233  isfin4p1  10237