Theorem fin4en1 9014

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 7891	. 2 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
2		bren 7850	. . . 4 ⊢ (𝐵 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐴)
3		simpr 476	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → 𝑥 ⊊ 𝐵)
4		f1of1 6049	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵–1-1→𝐴)
5		pssss 3664	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊊ 𝐵 → 𝑥 ⊆ 𝐵)
6		ssid 3587	. . . . . . . . . . . . . 14 ⊢ 𝐵 ⊆ 𝐵
7	5, 6	jctir 559	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊊ 𝐵 → (𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵))
8		f1imapss 6424	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ (𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵)) → ((𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵) ↔ 𝑥 ⊊ 𝐵))
9	4, 7, 8	syl2an 493	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → ((𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵) ↔ 𝑥 ⊊ 𝐵))
10	3, 9	mpbird 246	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵))
11		imadmrn 5395	. . . . . . . . . . . . . 14 ⊢ (𝑓 “ dom 𝑓) = ran 𝑓
12		f1odm 6054	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → dom 𝑓 = 𝐵)
13	12	imaeq2d 5385	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝑓 “ dom 𝑓) = (𝑓 “ 𝐵))
14		dff1o5 6059	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 ↔ (𝑓:𝐵–1-1→𝐴 ∧ ran 𝑓 = 𝐴))
15	14	simprbi 479	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → ran 𝑓 = 𝐴)
16	11, 13, 15	3eqtr3a 2668	. . . . . . . . . . . . 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 3662	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → ((𝑓 “ 𝑥) ⊊ (𝑓 “ 𝐵) ↔ (𝑓 “ 𝑥) ⊊ 𝐴))
19	10, 18	mpbid 221	. . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝑥) ⊊ 𝐴)
20	19	adantrr 749	. . . . . . . . 9 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ⊊ 𝐴)
21		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
22	21	f1imaen 7904	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝑥) ≈ 𝑥)
23	4, 5, 22	syl2an 493	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝑥 ⊊ 𝐵) → (𝑓 “ 𝑥) ≈ 𝑥)
24	23	adantrr 749	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝑥)
25		simprr 792	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → 𝑥 ≈ 𝐵)
26		entr 7894	. . . . . . . . . . 11 ⊢ (((𝑓 “ 𝑥) ≈ 𝑥 ∧ 𝑥 ≈ 𝐵) → (𝑓 “ 𝑥) ≈ 𝐵)
27	24, 25, 26	syl2anc 691	. . . . . . . . . 10 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝐵)
28		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑓 ∈ V
29		f1oen3g 7857	. . . . . . . . . . . 12 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐵–1-1-onto→𝐴) → 𝐵 ≈ 𝐴)
30	28, 29	mpan 702	. . . . . . . . . . 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 7894	. . . . . . . . . 10 ⊢ (((𝑓 “ 𝑥) ≈ 𝐵 ∧ 𝐵 ≈ 𝐴) → (𝑓 “ 𝑥) ≈ 𝐴)
33	27, 31, 32	syl2anc 691	. . . . . . . . 9 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → (𝑓 “ 𝑥) ≈ 𝐴)
34		fin4i 9003	. . . . . . . . 9 ⊢ (((𝑓 “ 𝑥) ⊊ 𝐴 ∧ (𝑓 “ 𝑥) ≈ 𝐴) → ¬ 𝐴 ∈ FinIV)
35	20, 33, 34	syl2anc 691	. . . . . . . 8 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ (𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)) → ¬ 𝐴 ∈ FinIV)
36	35	ex 449	. . . . . . 7 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → ((𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵) → ¬ 𝐴 ∈ FinIV))
37	36	exlimdv 1848	. . . . . 6 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵) → ¬ 𝐴 ∈ FinIV))
38	37	con2d 128	. . . . 5 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
39	38	exlimiv 1845	. . . 4 ⊢ (∃𝑓 𝑓:𝐵–1-1-onto→𝐴 → (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
40	2, 39	sylbi 206	. . 3 ⊢ (𝐵 ≈ 𝐴 → (𝐴 ∈ FinIV → ¬ ∃𝑥(𝑥 ⊊ 𝐵 ∧ 𝑥 ≈ 𝐵)))
41		relen 7846	. . . . 5 ⊢ Rel ≈
42	41	brrelexi 5082	. . . 4 ⊢ (𝐵 ≈ 𝐴 → 𝐵 ∈ V)
43		isfin4 9002	. . . 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 248	. 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 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540   ⊊ wpss 3541   class class class wbr 4583  dom cdm 5038  ran crn 5039   “ cima 5041  –1-1→wf1 5801  –1-1-onto→wf1o 5803   ≈ cen 7838  Fin^IVcfin4 8985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-er 7629  df-en 7842  df-fin4 8992

This theorem is referenced by:  domfin4  9016  isfin4-3  9020