Theorem enfin1ai 10320

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 8943	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
2		bren 8893	. . 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 4567	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐵 → 𝑥 ⊆ 𝐵)
5		simplr 767	. . . . . . . . 9 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝐴 ∈ FinIa)
6		imassrn 6024	. . . . . . . . . 10 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓
7		f1of 6784	. . . . . . . . . . . 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 6676	. . . . . . . . . 10 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → ran 𝑓 ⊆ 𝐴)
10	6, 9	sstrid 3955	. . . . . . . . 9 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝑥) ⊆ 𝐴)
11		fin1ai 10229	. . . . . . . . 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 6783	. . . . . . . . . . . 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 3449	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
17	16	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑥 ∈ V)
18		f1imaeng 8954	. . . . . . . . . . 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 9134	. . . . . . . . . 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 6501	. . . . . . . . . . . . . 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 6585	. . . . . . . . . . . . 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 6791	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵–onto→𝐴)
27		foima 6761	. . . . . . . . . . . . . . 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 4081	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → ((𝑓 “ 𝐵) ∖ (𝑓 “ 𝑥)) = (𝐴 ∖ (𝑓 “ 𝑥)))
31	25, 30	eqtrd 2776	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ (𝐵 ∖ 𝑥)) = (𝐴 ∖ (𝑓 “ 𝑥)))
32		difssd 4092	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐵 ∖ 𝑥) ⊆ 𝐵)
33		vex 3449	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
34	7	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → 𝑓:𝐵⟶𝐴)
35		dmfex 7844	. . . . . . . . . . . . . . 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 5286	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐵 ∖ 𝑥) ∈ V)
39		f1imaeng 8954	. . . . . . . . . . . 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 5129	. . . . . . . . . 10 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐴 ∖ (𝑓 “ 𝑥)) ≈ (𝐵 ∖ 𝑥))
42		enfi 9134	. . . . . . . . . 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 3143	. . . . 5 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → ∀𝑥 ∈ 𝒫 𝐵(𝑥 ∈ Fin ∨ (𝐵 ∖ 𝑥) ∈ Fin))
48		isfin1a 10228	. . . . . 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 3064  Vcvv 3445   ∖ cdif 3907   ⊆ wss 3910  𝒫 cpw 4560   class class class wbr 5105  ^◡ccnv 5632  ran crn 5634   “ cima 5636  Fun wfun 6490  ⟶wf 6492  –1-1→wf1 6493  –onto→wfo 6494  –1-1-onto→wf1o 6495   ≈ cen 8880  Fincfn 8883  Fin^Iacfin1a 10214

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-er 8648  df-en 8884  df-fin 8887  df-fin1a 10221

This theorem is referenced by: (None)