Theorem enfin1ai 10381

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 9001	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
2		bren 8951	. . 3 ⊢ (𝐵 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐴)
3	1, 2	sylib 217	. 2 ⊢ (𝐴 ≈ 𝐵 → ∃𝑓 𝑓:𝐵–1-1-onto→𝐴)
4		elpwi 4609	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐵 → 𝑥 ⊆ 𝐵)
5		simplr 767	. . . . . . . . 9 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝐴 ∈ FinIa)
6		imassrn 6070	. . . . . . . . . 10 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓
7		f1of 6833	. . . . . . . . . . . 12 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵⟶𝐴)
8	7	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑓:𝐵⟶𝐴)
9	8	frnd 6725	. . . . . . . . . 10 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → ran 𝑓 ⊆ 𝐴)
10	6, 9	sstrid 3993	. . . . . . . . 9 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝑥) ⊆ 𝐴)
11		fin1ai 10290	. . . . . . . . 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 6832	. . . . . . . . . . . 12 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵–1-1→𝐴)
14	13	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑓:𝐵–1-1→𝐴)
15		simpr 485	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ 𝐵)
16		vex 3478	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
17	16	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑥 ∈ V)
18		f1imaeng 9012	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑥 ∈ V) → (𝑓 “ 𝑥) ≈ 𝑥)
19	14, 15, 17, 18	syl3anc 1371	. . . . . . . . . 10 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝑥) ≈ 𝑥)
20		enfi 9192	. . . . . . . . . 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 6548	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1→𝐴 ↔ (𝑓:𝐵⟶𝐴 ∧ Fun ◡𝑓))
23	22	simprbi 497	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐵–1-1→𝐴 → Fun ◡𝑓)
24		imadif 6632	. . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵–onto→𝐴)
27		foima 6810	. . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝐵) = 𝐴)
30	29	difeq1d 4121	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → ((𝑓 “ 𝐵) ∖ (𝑓 “ 𝑥)) = (𝐴 ∖ (𝑓 “ 𝑥)))
31	25, 30	eqtrd 2772	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ (𝐵 ∖ 𝑥)) = (𝐴 ∖ (𝑓 “ 𝑥)))
32		difssd 4132	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐵 ∖ 𝑥) ⊆ 𝐵)
33		vex 3478	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
34	7	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → 𝑓:𝐵⟶𝐴)
35		dmfex 7900	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝐵 ∈ V)
38	37	difexd 5329	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐵 ∖ 𝑥) ∈ V)
39		f1imaeng 9012	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ (𝐵 ∖ 𝑥) ⊆ 𝐵 ∧ (𝐵 ∖ 𝑥) ∈ V) → (𝑓 “ (𝐵 ∖ 𝑥)) ≈ (𝐵 ∖ 𝑥))
40	14, 32, 38, 39	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ (𝐵 ∖ 𝑥)) ≈ (𝐵 ∖ 𝑥))
41	31, 40	eqbrtrrd 5172	. . . . . . . . . 10 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐴 ∖ (𝑓 “ 𝑥)) ≈ (𝐵 ∖ 𝑥))
42		enfi 9192	. . . . . . . . . 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 917	. . . . . . . 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 231	. . . . . . 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 3146	. . . . 5 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → ∀𝑥 ∈ 𝒫 𝐵(𝑥 ∈ Fin ∨ (𝐵 ∖ 𝑥) ∈ Fin))
48		isfin1a 10289	. . . . . 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 256	. . . 4 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → 𝐵 ∈ FinIa)
51	50	ex 413	. . 3 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝐴 ∈ FinIa → 𝐵 ∈ FinIa))
52	51	exlimiv 1933	. 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 205   ∧ wa 396   ∨ wo 845   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ∖ cdif 3945   ⊆ wss 3948  𝒫 cpw 4602   class class class wbr 5148  ^◡ccnv 5675  ran crn 5677   “ cima 5679  Fun wfun 6537  ⟶wf 6539  –1-1→wf1 6540  –onto→wfo 6541  –1-1-onto→wf1o 6542   ≈ cen 8938  Fincfn 8941  Fin^Iacfin1a 10275

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-fin 8945  df-fin1a 10282

This theorem is referenced by: (None)