Theorem enfin1ai 10301

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 8944	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴)
2		bren 8897	. . 3 ⊢ (𝐵 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝐴)
3	1, 2	sylib 220	. 2 ⊢ (𝐴 ≈ 𝐵 → ∃𝑓 𝑓:𝐵–1-1-onto→𝐴)
4		elpwi 4539	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝐵 → 𝑥 ⊆ 𝐵)
5		simplr 775	. . . . . . . . 9 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝐴 ∈ FinIa)
6		imassrn 6030	. . . . . . . . . 10 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓
7		f1of 6771	. . . . . . . . . . . 12 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵⟶𝐴)
8	7	ad2antrr 733	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑓:𝐵⟶𝐴)
9	8	frnd 6667	. . . . . . . . . 10 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → ran 𝑓 ⊆ 𝐴)
10	6, 9	sstrid 3928	. . . . . . . . 9 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝑥) ⊆ 𝐴)
11		fin1ai 10210	. . . . . . . . 9 ⊢ ((𝐴 ∈ FinIa ∧ (𝑓 “ 𝑥) ⊆ 𝐴) → ((𝑓 “ 𝑥) ∈ Fin ∨ (𝐴 ∖ (𝑓 “ 𝑥)) ∈ Fin))
12	5, 10, 11	syl2anc 591	. . . . . . . 8 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → ((𝑓 “ 𝑥) ∈ Fin ∨ (𝐴 ∖ (𝑓 “ 𝑥)) ∈ Fin))
13		f1of1 6770	. . . . . . . . . . . 12 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵–1-1→𝐴)
14	13	ad2antrr 733	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑓:𝐵–1-1→𝐴)
15		simpr 486	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑥 ⊆ 𝐵)
16		vex 3437	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
17	16	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝑥 ∈ V)
18		f1imaeng 8955	. . . . . . . . . . 11 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ 𝑥 ⊆ 𝐵 ∧ 𝑥 ∈ V) → (𝑓 “ 𝑥) ≈ 𝑥)
19	14, 15, 17, 18	syl3anc 1380	. . . . . . . . . 10 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝑥) ≈ 𝑥)
20		enfi 9115	. . . . . . . . . 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 6494	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐵–1-1→𝐴 ↔ (𝑓:𝐵⟶𝐴 ∧ Fun ◡𝑓))
23	22	simprbi 499	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐵–1-1→𝐴 → Fun ◡𝑓)
24		imadif 6573	. . . . . . . . . . . . 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 6778	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → 𝑓:𝐵–onto→𝐴)
27		foima 6748	. . . . . . . . . . . . . . 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 733	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ 𝐵) = 𝐴)
30	29	difeq1d 4059	. . . . . . . . . . . 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 4070	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐵 ∖ 𝑥) ⊆ 𝐵)
33		vex 3437	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
34	7	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → 𝑓:𝐵⟶𝐴)
35		dmfex 7849	. . . . . . . . . . . . . . 15 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐵⟶𝐴) → 𝐵 ∈ V)
36	33, 34, 35	sylancr 594	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → 𝐵 ∈ V)
37	36	adantr 482	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → 𝐵 ∈ V)
38	37	difexd 5262	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐵 ∖ 𝑥) ∈ V)
39		f1imaeng 8955	. . . . . . . . . . . 12 ⊢ ((𝑓:𝐵–1-1→𝐴 ∧ (𝐵 ∖ 𝑥) ⊆ 𝐵 ∧ (𝐵 ∖ 𝑥) ∈ V) → (𝑓 “ (𝐵 ∖ 𝑥)) ≈ (𝐵 ∖ 𝑥))
40	14, 32, 38, 39	syl3anc 1380	. . . . . . . . . . 11 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑓 “ (𝐵 ∖ 𝑥)) ≈ (𝐵 ∖ 𝑥))
41	31, 40	eqbrtrrd 5099	. . . . . . . . . 10 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝐴 ∖ (𝑓 “ 𝑥)) ≈ (𝐵 ∖ 𝑥))
42		enfi 9115	. . . . . . . . . 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 925	. . . . . . . 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 234	. . . . . . 7 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ⊆ 𝐵) → (𝑥 ∈ Fin ∨ (𝐵 ∖ 𝑥) ∈ Fin))
46	4, 45	sylan2 600	. . . . . 6 ⊢ (((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) ∧ 𝑥 ∈ 𝒫 𝐵) → (𝑥 ∈ Fin ∨ (𝐵 ∖ 𝑥) ∈ Fin))
47	46	ralrimiva 3133	. . . . 5 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → ∀𝑥 ∈ 𝒫 𝐵(𝑥 ∈ Fin ∨ (𝐵 ∖ 𝑥) ∈ Fin))
48		isfin1a 10209	. . . . . 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 259	. . . 4 ⊢ ((𝑓:𝐵–1-1-onto→𝐴 ∧ 𝐴 ∈ FinIa) → 𝐵 ∈ FinIa)
51	50	ex 414	. . 3 ⊢ (𝑓:𝐵–1-1-onto→𝐴 → (𝐴 ∈ FinIa → 𝐵 ∈ FinIa))
52	51	exlimiv 1938	. 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 208   ∧ wa 397   ∨ wo 854   = wceq 1548  ∃wex 1787   ∈ wcel 2121  ∀wral 3055  Vcvv 3433   ∖ cdif 3882   ⊆ wss 3885  𝒫 cpw 4532   class class class wbr 5075  ^◡ccnv 5620  ran crn 5622   “ cima 5624  Fun wfun 6483  ⟶wf 6485  –1-1→wf1 6486  –onto→wfo 6487  –1-1-onto→wf1o 6488   ≈ cen 8884  Fincfn 8887  Fin^Iacfin1a 10195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-om 7811  df-1o 8399  df-er 8637  df-en 8888  df-fin 8891  df-fin1a 10202

This theorem is referenced by: (None)