Theorem enfin2i 10231

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

Assertion

Ref	Expression
enfin2i	⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinII → 𝐵 ∈ FinII))

Proof of Theorem enfin2i

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

Step	Hyp	Ref	Expression
1		bren 8893	. . 3 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		elpwi 4561	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐵 → 𝑥 ⊆ 𝒫 𝐵)
3		imauni 7192	. . . . . . . . . . 11 ⊢ (𝑓 “ ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}) = ∪ 𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} (𝑓 “ 𝑧)
4		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑓 ∈ V
5	4	imaex 7856	. . . . . . . . . . . 12 ⊢ (𝑓 “ 𝑧) ∈ V
6	5	dfiun2 4987	. . . . . . . . . . 11 ⊢ ∪ 𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} (𝑓 “ 𝑧) = ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}𝑤 = (𝑓 “ 𝑧)}
7	3, 6	eqtri 2759	. . . . . . . . . 10 ⊢ (𝑓 “ ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}) = ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}𝑤 = (𝑓 “ 𝑧)}
8		imaeq2 6015	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝑓 “ 𝑦) = (𝑓 “ 𝑧))
9	8	eleq1d 2821	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → ((𝑓 “ 𝑦) ∈ 𝑥 ↔ (𝑓 “ 𝑧) ∈ 𝑥))
10	9	rexrab 3654	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}𝑤 = (𝑓 “ 𝑧) ↔ ∃𝑧 ∈ 𝒫 𝐴((𝑓 “ 𝑧) ∈ 𝑥 ∧ 𝑤 = (𝑓 “ 𝑧)))
11		eleq1 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓 “ 𝑧) → (𝑤 ∈ 𝑥 ↔ (𝑓 “ 𝑧) ∈ 𝑥))
12	11	biimparc 479	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 “ 𝑧) ∈ 𝑥 ∧ 𝑤 = (𝑓 “ 𝑧)) → 𝑤 ∈ 𝑥)
13	12	rexlimivw 3133	. . . . . . . . . . . . . 14 ⊢ (∃𝑧 ∈ 𝒫 𝐴((𝑓 “ 𝑧) ∈ 𝑥 ∧ 𝑤 = (𝑓 “ 𝑧)) → 𝑤 ∈ 𝑥)
14		cnvimass 6041	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑓 “ 𝑤) ⊆ dom 𝑓
15		f1odm 6778	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
16	15	ad3antrrr 730	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → dom 𝑓 = 𝐴)
17	14, 16	sseqtrid 3976	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → (◡𝑓 “ 𝑤) ⊆ 𝐴)
18	4	cnvex 7867	. . . . . . . . . . . . . . . . . . 19 ⊢ ◡𝑓 ∈ V
19	18	imaex 7856	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑓 “ 𝑤) ∈ V
20	19	elpw 4558	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑓 “ 𝑤) ∈ 𝒫 𝐴 ↔ (◡𝑓 “ 𝑤) ⊆ 𝐴)
21	17, 20	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → (◡𝑓 “ 𝑤) ∈ 𝒫 𝐴)
22		f1ofo 6781	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
23	22	ad3antrrr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → 𝑓:𝐴–onto→𝐵)
24		simprl 770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → 𝑥 ⊆ 𝒫 𝐵)
25	24	sselda 3933	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ 𝒫 𝐵)
26	25	elpwid 4563	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → 𝑤 ⊆ 𝐵)
27		foimacnv 6791	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝐴–onto→𝐵 ∧ 𝑤 ⊆ 𝐵) → (𝑓 “ (◡𝑓 “ 𝑤)) = 𝑤)
28	23, 26, 27	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → (𝑓 “ (◡𝑓 “ 𝑤)) = 𝑤)
29	28	eqcomd 2742	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → 𝑤 = (𝑓 “ (◡𝑓 “ 𝑤)))
30		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ 𝑥)
31	29, 30	eqeltrrd 2837	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → (𝑓 “ (◡𝑓 “ 𝑤)) ∈ 𝑥)
32		imaeq2 6015	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (◡𝑓 “ 𝑤) → (𝑓 “ 𝑧) = (𝑓 “ (◡𝑓 “ 𝑤)))
33	32	eleq1d 2821	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (◡𝑓 “ 𝑤) → ((𝑓 “ 𝑧) ∈ 𝑥 ↔ (𝑓 “ (◡𝑓 “ 𝑤)) ∈ 𝑥))
34	32	eqeq2d 2747	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (◡𝑓 “ 𝑤) → (𝑤 = (𝑓 “ 𝑧) ↔ 𝑤 = (𝑓 “ (◡𝑓 “ 𝑤))))
35	33, 34	anbi12d 632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (◡𝑓 “ 𝑤) → (((𝑓 “ 𝑧) ∈ 𝑥 ∧ 𝑤 = (𝑓 “ 𝑧)) ↔ ((𝑓 “ (◡𝑓 “ 𝑤)) ∈ 𝑥 ∧ 𝑤 = (𝑓 “ (◡𝑓 “ 𝑤)))))
36	35	rspcev 3576	. . . . . . . . . . . . . . . 16 ⊢ (((◡𝑓 “ 𝑤) ∈ 𝒫 𝐴 ∧ ((𝑓 “ (◡𝑓 “ 𝑤)) ∈ 𝑥 ∧ 𝑤 = (𝑓 “ (◡𝑓 “ 𝑤)))) → ∃𝑧 ∈ 𝒫 𝐴((𝑓 “ 𝑧) ∈ 𝑥 ∧ 𝑤 = (𝑓 “ 𝑧)))
37	21, 31, 29, 36	syl12anc 836	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → ∃𝑧 ∈ 𝒫 𝐴((𝑓 “ 𝑧) ∈ 𝑥 ∧ 𝑤 = (𝑓 “ 𝑧)))
38	37	ex 412	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → (𝑤 ∈ 𝑥 → ∃𝑧 ∈ 𝒫 𝐴((𝑓 “ 𝑧) ∈ 𝑥 ∧ 𝑤 = (𝑓 “ 𝑧))))
39	13, 38	impbid2 226	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → (∃𝑧 ∈ 𝒫 𝐴((𝑓 “ 𝑧) ∈ 𝑥 ∧ 𝑤 = (𝑓 “ 𝑧)) ↔ 𝑤 ∈ 𝑥))
40	10, 39	bitrid 283	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → (∃𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}𝑤 = (𝑓 “ 𝑧) ↔ 𝑤 ∈ 𝑥))
41	40	eqabcdv 2870	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}𝑤 = (𝑓 “ 𝑧)} = 𝑥)
42	41	unieqd 4876	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}𝑤 = (𝑓 “ 𝑧)} = ∪ 𝑥)
43	7, 42	eqtrid 2783	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → (𝑓 “ ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}) = ∪ 𝑥)
44		simplr 768	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → 𝐴 ∈ FinII)
45		ssrab2 4032	. . . . . . . . . . . 12 ⊢ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ⊆ 𝒫 𝐴
46	45	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ⊆ 𝒫 𝐴)
47		simprrl 780	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → 𝑥 ≠ ∅)
48		n0 4305	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝑥)
49	47, 48	sylib 218	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → ∃𝑤 𝑤 ∈ 𝑥)
50		imaeq2 6015	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (◡𝑓 “ 𝑤) → (𝑓 “ 𝑦) = (𝑓 “ (◡𝑓 “ 𝑤)))
51	50	eleq1d 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (◡𝑓 “ 𝑤) → ((𝑓 “ 𝑦) ∈ 𝑥 ↔ (𝑓 “ (◡𝑓 “ 𝑤)) ∈ 𝑥))
52	51	rspcev 3576	. . . . . . . . . . . . . 14 ⊢ (((◡𝑓 “ 𝑤) ∈ 𝒫 𝐴 ∧ (𝑓 “ (◡𝑓 “ 𝑤)) ∈ 𝑥) → ∃𝑦 ∈ 𝒫 𝐴(𝑓 “ 𝑦) ∈ 𝑥)
53	21, 31, 52	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → ∃𝑦 ∈ 𝒫 𝐴(𝑓 “ 𝑦) ∈ 𝑥)
54	49, 53	exlimddv 1936	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → ∃𝑦 ∈ 𝒫 𝐴(𝑓 “ 𝑦) ∈ 𝑥)
55		rabn0 4341	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ≠ ∅ ↔ ∃𝑦 ∈ 𝒫 𝐴(𝑓 “ 𝑦) ∈ 𝑥)
56	54, 55	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ≠ ∅)
57	9	elrab 3646	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥))
58		imaeq2 6015	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑤 → (𝑓 “ 𝑦) = (𝑓 “ 𝑤))
59	58	eleq1d 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → ((𝑓 “ 𝑦) ∈ 𝑥 ↔ (𝑓 “ 𝑤) ∈ 𝑥))
60	59	elrab 3646	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ↔ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))
61	57, 60	anbi12i 628	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ∧ 𝑤 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}) ↔ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥)))
62		simprrr 781	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → [⊊] Or 𝑥)
63	62	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))) → [⊊] Or 𝑥)
64		simprlr 779	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))) → (𝑓 “ 𝑧) ∈ 𝑥)
65		simprrr 781	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))) → (𝑓 “ 𝑤) ∈ 𝑥)
66		sorpssi 7674	. . . . . . . . . . . . . . . 16 ⊢ (( [⊊] Or 𝑥 ∧ ((𝑓 “ 𝑧) ∈ 𝑥 ∧ (𝑓 “ 𝑤) ∈ 𝑥)) → ((𝑓 “ 𝑧) ⊆ (𝑓 “ 𝑤) ∨ (𝑓 “ 𝑤) ⊆ (𝑓 “ 𝑧)))
67	63, 64, 65, 66	syl12anc 836	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))) → ((𝑓 “ 𝑧) ⊆ (𝑓 “ 𝑤) ∨ (𝑓 “ 𝑤) ⊆ (𝑓 “ 𝑧)))
68		f1of1 6773	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–1-1→𝐵)
69	68	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))) → 𝑓:𝐴–1-1→𝐵)
70		simprll 778	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))) → 𝑧 ∈ 𝒫 𝐴)
71	70	elpwid 4563	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))) → 𝑧 ⊆ 𝐴)
72		simprrl 780	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))) → 𝑤 ∈ 𝒫 𝐴)
73	72	elpwid 4563	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))) → 𝑤 ⊆ 𝐴)
74		f1imass 7210	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ⊆ 𝐴)) → ((𝑓 “ 𝑧) ⊆ (𝑓 “ 𝑤) ↔ 𝑧 ⊆ 𝑤))
75	69, 71, 73, 74	syl12anc 836	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))) → ((𝑓 “ 𝑧) ⊆ (𝑓 “ 𝑤) ↔ 𝑧 ⊆ 𝑤))
76		f1imass 7210	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ (𝑤 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴)) → ((𝑓 “ 𝑤) ⊆ (𝑓 “ 𝑧) ↔ 𝑤 ⊆ 𝑧))
77	69, 73, 71, 76	syl12anc 836	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))) → ((𝑓 “ 𝑤) ⊆ (𝑓 “ 𝑧) ↔ 𝑤 ⊆ 𝑧))
78	75, 77	orbi12d 918	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))) → (((𝑓 “ 𝑧) ⊆ (𝑓 “ 𝑤) ∨ (𝑓 “ 𝑤) ⊆ (𝑓 “ 𝑧)) ↔ (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧)))
79	67, 78	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ ((𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))) → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))
80	61, 79	sylan2b 594	. . . . . . . . . . . . 13 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ∧ 𝑤 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥})) → (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))
81	80	ralrimivva 3179	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → ∀𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}∀𝑤 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))
82		sorpss 7673	. . . . . . . . . . . 12 ⊢ ( [⊊] Or {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ↔ ∀𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}∀𝑤 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} (𝑧 ⊆ 𝑤 ∨ 𝑤 ⊆ 𝑧))
83	81, 82	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → [⊊] Or {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥})
84		fin2i 10205	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ FinII ∧ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ⊆ 𝒫 𝐴) ∧ ({𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ≠ ∅ ∧ [⊊] Or {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥})) → ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥})
85	44, 46, 56, 83, 84	syl22anc 838	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥})
86		imaeq2 6015	. . . . . . . . . . . . 13 ⊢ (𝑧 = ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} → (𝑓 “ 𝑧) = (𝑓 “ ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}))
87	86	eleq1d 2821	. . . . . . . . . . . 12 ⊢ (𝑧 = ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} → ((𝑓 “ 𝑧) ∈ 𝑥 ↔ (𝑓 “ ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}) ∈ 𝑥))
88	9	cbvrabv 3409	. . . . . . . . . . . 12 ⊢ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} = {𝑧 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑧) ∈ 𝑥}
89	87, 88	elrab2 3649	. . . . . . . . . . 11 ⊢ (∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ↔ (∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ∈ 𝒫 𝐴 ∧ (𝑓 “ ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}) ∈ 𝑥))
90	89	simprbi 496	. . . . . . . . . 10 ⊢ (∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} → (𝑓 “ ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}) ∈ 𝑥)
91	85, 90	syl 17	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → (𝑓 “ ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}) ∈ 𝑥)
92	43, 91	eqeltrrd 2837	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → ∪ 𝑥 ∈ 𝑥)
93	92	expr 456	. . . . . . 7 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ 𝑥 ⊆ 𝒫 𝐵) → ((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))
94	2, 93	sylan2 593	. . . . . 6 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → ((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))
95	94	ralrimiva 3128	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))
96	95	ex 412	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ FinII → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)))
97	96	exlimiv 1931	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ FinII → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)))
98	1, 97	sylbi 217	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinII → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)))
99		relen 8888	. . . 4 ⊢ Rel ≈
100	99	brrelex2i 5681	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
101		isfin2 10204	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ FinII ↔ ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)))
102	100, 101	syl 17	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐵 ∈ FinII ↔ ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)))
103	98, 102	sylibrd 259	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinII → 𝐵 ∈ FinII))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3399  Vcvv 3440   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  ∪ cuni 4863  ∪ ciun 4946   class class class wbr 5098   Or wor 5531  ^◡ccnv 5623  dom cdm 5624   “ cima 5627  –1-1→wf1 6489  –onto→wfo 6490  –1-1-onto→wf1o 6491   [⊊] crpss 7667   ≈ cen 8880  Fin^IIcfin2 10189

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-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-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  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-id 5519  df-po 5532  df-so 5533  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-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-rpss 7668  df-en 8884  df-fin2 10196

This theorem is referenced by: (None)