Theorem enfin1ai 10296

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 8942	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
2		bren 8895	. . 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 4561	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐵 → 𝑥 ⊆ 𝐵)
5		simplr 768	. . . . . . . . 9 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝐴 ∈ FinIa)
6		imassrn 6030	. . . . . . . . . 10 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓
7		f1of 6774	. . . . . . . . . . . 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 6670	. . . . . . . . . 10 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → ran 𝑓 ⊆ 𝐴)
10	6, 9	sstrid 3945	. . . . . . . . 9 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝑥) ⊆ 𝐴)
11		fin1ai 10205	. . . . . . . . 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 6773	. . . . . . . . . . . 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 3444	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
17	16	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑥 ∈ V)
18		f1imaeng 8953	. . . . . . . . . . 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 9113	. . . . . . . . . 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 6497	. . . . . . . . . . . . . 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 6576	. . . . . . . . . . . . 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 6781	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵–onto→𝐴)
27		foima 6751	. . . . . . . . . . . . . . 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 4077	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → ((𝑓 “ 𝐵) ∖ (𝑓 “ 𝑥)) = (𝐴 ∖ (𝑓 “ 𝑥)))
31	25, 30	eqtrd 2771	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ (𝐵 ∖ 𝑥)) = (𝐴 ∖ (𝑓 “ 𝑥)))
32		difssd 4089	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐵 ∖ 𝑥) ⊆ 𝐵)
33		vex 3444	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
34	7	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → 𝑓:𝐵⟶𝐴)
35		dmfex 7847	. . . . . . . . . . . . . . 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 5276	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐵 ∖ 𝑥) ∈ V)
39		f1imaeng 8953	. . . . . . . . . . . 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 5122	. . . . . . . . . 10 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐴 ∖ (𝑓 “ 𝑥)) ≈ (𝐵 ∖ 𝑥))
42		enfi 9113	. . . . . . . . . 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 3128	. . . . 5 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → ∀𝑥 ∈ 𝒫 𝐵(𝑥 ∈ Fin ∨ (𝐵 ∖ 𝑥) ∈ Fin))
48		isfin1a 10204	. . . . . 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 1931	. 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 1541  ∃wex 1780   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901  𝒫 cpw 4554   class class class wbr 5098  ^◡ccnv 5623  ran crn 5625   “ cima 5627  Fun wfun 6486  ⟶wf 6488  –1-1→wf1 6489  –onto→wfo 6490  –1-1-onto→wf1o 6491   ≈ cen 8882  Fincfn 8885  Fin^Iacfin1a 10190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-er 8635  df-en 8886  df-fin 8889  df-fin1a 10197

This theorem is referenced by: (None)