Theorem enfin1ai 10337

Description: Ia-finiteness is a cardinal property. (Contributed by Mario Carneiro, 18-May-2015.)

Assertion

Ref	Expression
enfin1ai	⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinIa → 𝐵 ∈ FinIa))

Proof of Theorem enfin1ai

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

Step	Hyp	Ref	Expression
1		ensym 8974	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
2		bren 8928	. . 3 ⊢ (𝐵 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐴)
3	1, 2	sylib 218	. 2 ⊢ (𝐴 ≈ 𝐵 → ∃𝑓 𝑓:𝐵–1-1-onto→𝐴)
4		elpwi 4570	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐵 → 𝑥 ⊆ 𝐵)
5		simplr 768	. . . . . . . . 9 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝐴 ∈ FinIa)
6		imassrn 6042	. . . . . . . . . 10 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓
7		f1of 6800	. . . . . . . . . . . 12 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵⟶𝐴)
8	7	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑓:𝐵⟶𝐴)
9	8	frnd 6696	. . . . . . . . . 10 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → ran 𝑓 ⊆ 𝐴)
10	6, 9	sstrid 3958	. . . . . . . . 9 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝑥) ⊆ 𝐴)
11		fin1ai 10246	. . . . . . . . 9 ⊢ ((𝐴 ∈ FinIa ∧ (𝑓 “ 𝑥) ⊆ 𝐴) → ((𝑓 “ 𝑥) ∈ Fin ∨ (𝐴 ∖ (𝑓 “ 𝑥)) ∈ Fin))
12	5, 10, 11	syl2anc 584	. . . . . . . 8 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → ((𝑓 “ 𝑥) ∈ Fin ∨ (𝐴 ∖ (𝑓 “ 𝑥)) ∈ Fin))
13		f1of1 6799	. . . . . . . . . . . 12 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵–1-1→𝐴)
14	13	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑓:𝐵–1-1→𝐴)
15		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ 𝐵)
16		vex 3451	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
17	16	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑥 ∈ V)
18		f1imaeng 8985	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑥 ∈ V) → (𝑓 “ 𝑥) ≈ 𝑥)
19	14, 15, 17, 18	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝑥) ≈ 𝑥)
20		enfi 9151	. . . . . . . . . 10 ⊢ ((𝑓 “ 𝑥) ≈ 𝑥 → ((𝑓 “ 𝑥) ∈ Fin ↔ 𝑥 ∈ Fin))
21	19, 20	syl 17	. . . . . . . . 9 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → ((𝑓 “ 𝑥) ∈ Fin ↔ 𝑥 ∈ Fin))
22		df-f1 6516	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1→𝐴 ↔ (𝑓:𝐵⟶𝐴 ∧ Fun ◡𝑓))
23	22	simprbi 496	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐵–1-1→𝐴 → Fun ◡𝑓)
24		imadif 6600	. . . . . . . . . . . . 13 ⊢ (Fun ◡𝑓 → (𝑓 “ (𝐵 ∖ 𝑥)) = ((𝑓 “ 𝐵) ∖ (𝑓 “ 𝑥)))
25	14, 23, 24	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ (𝐵 ∖ 𝑥)) = ((𝑓 “ 𝐵) ∖ (𝑓 “ 𝑥)))
26		f1ofo 6807	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵–onto→𝐴)
27		foima 6777	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–onto→𝐴 → (𝑓 “ 𝐵) = 𝐴)
28	26, 27	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝑓 “ 𝐵) = 𝐴)
29	28	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝐵) = 𝐴)
30	29	difeq1d 4088	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → ((𝑓 “ 𝐵) ∖ (𝑓 “ 𝑥)) = (𝐴 ∖ (𝑓 “ 𝑥)))
31	25, 30	eqtrd 2764	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ (𝐵 ∖ 𝑥)) = (𝐴 ∖ (𝑓 “ 𝑥)))
32		difssd 4100	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐵 ∖ 𝑥) ⊆ 𝐵)
33		vex 3451	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
34	7	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → 𝑓:𝐵⟶𝐴)
35		dmfex 7881	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐵⟶𝐴) → 𝐵 ∈ V)
36	33, 34, 35	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → 𝐵 ∈ V)
37	36	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝐵 ∈ V)
38	37	difexd 5286	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐵 ∖ 𝑥) ∈ V)
39		f1imaeng 8985	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ (𝐵 ∖ 𝑥) ⊆ 𝐵 ∧ (𝐵 ∖ 𝑥) ∈ V) → (𝑓 “ (𝐵 ∖ 𝑥)) ≈ (𝐵 ∖ 𝑥))
40	14, 32, 38, 39	syl3anc 1373	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ (𝐵 ∖ 𝑥)) ≈ (𝐵 ∖ 𝑥))
41	31, 40	eqbrtrrd 5131	. . . . . . . . . 10 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐴 ∖ (𝑓 “ 𝑥)) ≈ (𝐵 ∖ 𝑥))
42		enfi 9151	. . . . . . . . . 10 ⊢ ((𝐴 ∖ (𝑓 “ 𝑥)) ≈ (𝐵 ∖ 𝑥) → ((𝐴 ∖ (𝑓 “ 𝑥)) ∈ Fin ↔ (𝐵 ∖ 𝑥) ∈ Fin))
43	41, 42	syl 17	. . . . . . . . 9 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → ((𝐴 ∖ (𝑓 “ 𝑥)) ∈ Fin ↔ (𝐵 ∖ 𝑥) ∈ Fin))
44	21, 43	orbi12d 918	. . . . . . . 8 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (((𝑓 “ 𝑥) ∈ Fin ∨ (𝐴 ∖ (𝑓 “ 𝑥)) ∈ Fin) ↔ (𝑥 ∈ Fin ∨ (𝐵 ∖ 𝑥) ∈ Fin)))
45	12, 44	mpbid 232	. . . . . . 7 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑥 ∈ Fin ∨ (𝐵 ∖ 𝑥) ∈ Fin))
46	4, 45	sylan2 593	. . . . . 6 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ∈ 𝒫 𝐵) → (𝑥 ∈ Fin ∨ (𝐵 ∖ 𝑥) ∈ Fin))
47	46	ralrimiva 3125	. . . . 5 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → ∀𝑥 ∈ 𝒫 𝐵(𝑥 ∈ Fin ∨ (𝐵 ∖ 𝑥) ∈ Fin))
48		isfin1a 10245	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 ∈ FinIa ↔ ∀𝑥 ∈ 𝒫 𝐵(𝑥 ∈ Fin ∨ (𝐵 ∖ 𝑥) ∈ Fin)))
49	36, 48	syl 17	. . . . 5 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → (𝐵 ∈ FinIa ↔ ∀𝑥 ∈ 𝒫 𝐵(𝑥 ∈ Fin ∨ (𝐵 ∖ 𝑥) ∈ Fin)))
50	47, 49	mpbird 257	. . . 4 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → 𝐵 ∈ FinIa)
51	50	ex 412	. . 3 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝐴 ∈ FinIa → 𝐵 ∈ FinIa))
52	51	exlimiv 1930	. 2 ⊢ (∃𝑓 𝑓:𝐵–1-1-onto→𝐴 → (𝐴 ∈ FinIa → 𝐵 ∈ FinIa))
53	3, 52	syl 17	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinIa → 𝐵 ∈ FinIa))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ∖ cdif 3911   ⊆ wss 3914  𝒫 cpw 4563   class class class wbr 5107  ^◡ccnv 5637  ran crn 5639   “ cima 5641  Fun wfun 6505  ⟶wf 6507  –1-1→wf1 6508  –onto→wfo 6509  –1-1-onto→wf1o 6510   ≈ cen 8915  Fincfn 8918  Fin^Iacfin1a 10231

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 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-1o 8434  df-er 8671  df-en 8919  df-fin 8922  df-fin1a 10238

This theorem is referenced by: (None)