Theorem enfin2i 10406

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 8990	. . 3 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		elpwi 4563	. . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐵 → 𝑥 ⊆ 𝒫 𝐵)
3		imauni 7254	. . . . . . . . . . 11 ⊢ (𝑓 “ ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}) = ∪ 𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} (𝑓 “ 𝑧)
4		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑓 ∈ V
5	4	imaex 7930	. . . . . . . . . . . 12 ⊢ (𝑓 “ 𝑧) ∈ V
6	5	dfiun2 4989	. . . . . . . . . . 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 2783	. . . . . . . . . 10 ⊢ (𝑓 “ ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}) = ∪ {𝑤 ∣ ∃𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}𝑤 = (𝑓 “ 𝑧)}
8		imaeq2 6050	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝑓 “ 𝑦) = (𝑓 “ 𝑧))
9	8	eleq1d 2845	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → ((𝑓 “ 𝑦) ∈ 𝑥 ↔ (𝑓 “ 𝑧) ∈ 𝑥))
10	9	rexrab 3653	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}𝑤 = (𝑓 “ 𝑧) ↔ ∃𝑧 ∈ 𝒫 𝐴((𝑓 “ 𝑧) ∈ 𝑥 ∧ 𝑤 = (𝑓 “ 𝑧)))
11		eleq1 2848	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = (𝑓 “ 𝑧) → (𝑤 ∈ 𝑥 ↔ (𝑓 “ 𝑧) ∈ 𝑥))
12	11	biimparc 485	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 “ 𝑧) ∈ 𝑥 ∧ 𝑤 = (𝑓 “ 𝑧)) → 𝑤 ∈ 𝑥)
13	12	rexlimivw 3159	. . . . . . . . . . . . . 14 ⊢ (∃𝑧 ∈ 𝒫 𝐴((𝑓 “ 𝑧) ∈ 𝑥 ∧ 𝑤 = (𝑓 “ 𝑧)) → 𝑤 ∈ 𝑥)
14		cnvimass 6201	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑓 “ 𝑤) ⊆ dom 𝑓
15		f1odm 6833	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
16	15	ad3antrrr 743	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → dom 𝑓 = 𝐴)
17	14, 16	sseqtrid 3951	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → (◡𝑓 “ 𝑤) ⊆ 𝐴)
18	4	cnvex 7941	. . . . . . . . . . . . . . . . . . 19 ⊢ ◡𝑓 ∈ V
19	18	imaex 7930	. . . . . . . . . . . . . . . . . 18 ⊢ (◡𝑓 “ 𝑤) ∈ V
20	19	elpw 4560	. . . . . . . . . . . . . . . . 17 ⊢ ((◡𝑓 “ 𝑤) ∈ 𝒫 𝐴 ↔ (◡𝑓 “ 𝑤) ⊆ 𝐴)
21	17, 20	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → (◡𝑓 “ 𝑤) ∈ 𝒫 𝐴)
22		f1ofo 6837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
23	22	ad3antrrr 743	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → 𝑓:𝐴–onto→𝐵)
24		simprl 783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → 𝑥 ⊆ 𝒫 𝐵)
25	24	sselda 3909	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ 𝒫 𝐵)
26	25	elpwid 4565	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → 𝑤 ⊆ 𝐵)
27		foimacnv 6847	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝐴–onto→𝐵 ∧ 𝑤 ⊆ 𝐵) → (𝑓 “ (◡𝑓 “ 𝑤)) = 𝑤)
28	23, 26, 27	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → (𝑓 “ (◡𝑓 “ 𝑤)) = 𝑤)
29	28	eqcomd 2766	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → 𝑤 = (𝑓 “ (◡𝑓 “ 𝑤)))
30		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → 𝑤 ∈ 𝑥)
31	29, 30	eqeltrrd 2861	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) ∧ 𝑤 ∈ 𝑥) → (𝑓 “ (◡𝑓 “ 𝑤)) ∈ 𝑥)
32		imaeq2 6050	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (◡𝑓 “ 𝑤) → (𝑓 “ 𝑧) = (𝑓 “ (◡𝑓 “ 𝑤)))
33	32	eleq1d 2845	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (◡𝑓 “ 𝑤) → ((𝑓 “ 𝑧) ∈ 𝑥 ↔ (𝑓 “ (◡𝑓 “ 𝑤)) ∈ 𝑥))
34	32	eqeq2d 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (◡𝑓 “ 𝑤) → (𝑤 = (𝑓 “ 𝑧) ↔ 𝑤 = (𝑓 “ (◡𝑓 “ 𝑤))))
35	33, 34	anbi12d 644	. . . . . . . . . . . . . . . . 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 850	. . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . 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 229	. . . . . . . . . . . . 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 286	. . . . . . . . . . . 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 2894	. . . . . . . . . . 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 4879	. . . . . . . . . 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 2807	. . . . . . . . 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 781	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → 𝐴 ∈ FinII)
45		ssrab2 4004	. . . . . . . . . . . 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 793	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → 𝑥 ≠ ∅)
48		n0 4299	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝑥)
49	47, 48	sylib 221	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → ∃𝑤 𝑤 ∈ 𝑥)
50		imaeq2 6050	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = (◡𝑓 “ 𝑤) → (𝑓 “ 𝑦) = (𝑓 “ (◡𝑓 “ 𝑤)))
51	50	eleq1d 2845	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 1967	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → ∃𝑦 ∈ 𝒫 𝐴(𝑓 “ 𝑦) ∈ 𝑥)
55		rabn0 4338	. . . . . . . . . . . 12 ⊢ ({𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ≠ ∅ ↔ ∃𝑦 ∈ 𝒫 𝐴(𝑓 “ 𝑦) ∈ 𝑥)
56	54, 55	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ≠ ∅)
57	9	elrab 3644	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑧) ∈ 𝑥))
58		imaeq2 6050	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑤 → (𝑓 “ 𝑦) = (𝑓 “ 𝑤))
59	58	eleq1d 2845	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → ((𝑓 “ 𝑦) ∈ 𝑥 ↔ (𝑓 “ 𝑤) ∈ 𝑥))
60	59	elrab 3644	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} ↔ (𝑤 ∈ 𝒫 𝐴 ∧ (𝑓 “ 𝑤) ∈ 𝑥))
61	57, 60	anbi12i 640	. . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → [⊊] Or 𝑥)
63	62	adantr 486	. . . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . 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 7749	. . . . . . . . . . . . . . . 16 ⊢ (( [⊊] Or 𝑥 ∧ ((𝑓 “ 𝑧) ∈ 𝑥 ∧ (𝑓 “ 𝑤) ∈ 𝑥)) → ((𝑓 “ 𝑧) ⊆ (𝑓 “ 𝑤) ∨ (𝑓 “ 𝑤) ⊆ (𝑓 “ 𝑧)))
67	63, 64, 65, 66	syl12anc 850	. . . . . . . . . . . . . . 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 6828	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–1-1→𝐵)
69	68	ad3antrrr 743	. . . . . . . . . . . . . . . . 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 791	. . . . . . . . . . . . . . . . . 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 4565	. . . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . . 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 4565	. . . . . . . . . . . . . . . . 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 7272	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ (𝑧 ⊆ 𝐴 ∧ 𝑤 ⊆ 𝐴)) → ((𝑓 “ 𝑧) ⊆ (𝑓 “ 𝑤) ↔ 𝑧 ⊆ 𝑤))
75	69, 71, 73, 74	syl12anc 850	. . . . . . . . . . . . . . . 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 7272	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ (𝑤 ⊆ 𝐴 ∧ 𝑧 ⊆ 𝐴)) → ((𝑓 “ 𝑤) ⊆ (𝑓 “ 𝑧) ↔ 𝑤 ⊆ 𝑧))
77	69, 73, 71, 76	syl12anc 850	. . . . . . . . . . . . . . . 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 932	. . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . 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 606	. . . . . . . . . . . . 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 3205	. . . . . . . . . . . 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 7748	. . . . . . . . . . . 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 237	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → [⊊] Or {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥})
84		fin2i 10380	. . . . . . . . . . 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 852	. . . . . . . . . 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 6050	. . . . . . . . . . . . 13 ⊢ (𝑧 = ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} → (𝑓 “ 𝑧) = (𝑓 “ ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}))
87	86	eleq1d 2845	. . . . . . . . . . . 12 ⊢ (𝑧 = ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} → ((𝑓 “ 𝑧) ∈ 𝑥 ↔ (𝑓 “ ∪ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥}) ∈ 𝑥))
88	9	cbvrabv 3422	. . . . . . . . . . . 12 ⊢ {𝑦 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑦) ∈ 𝑥} = {𝑧 ∈ 𝒫 𝐴 ∣ (𝑓 “ 𝑧) ∈ 𝑥}
89	87, 88	elrab2 3648	. . . . . . . . . . 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 503	. . . . . . . . . 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 18	. . . . . . . . 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 2861	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ (𝑥 ⊆ 𝒫 𝐵 ∧ (𝑥 ≠ ∅ ∧ [⊊] Or 𝑥))) → ∪ 𝑥 ∈ 𝑥)
93	92	expr 462	. . . . . . 7 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ 𝑥 ⊆ 𝒫 𝐵) → ((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))
94	2, 93	sylan2 605	. . . . . 6 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) ∧ 𝑥 ∈ 𝒫 𝒫 𝐵) → ((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))
95	94	ralrimiva 3154	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝐴 ∈ FinII) → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥))
96	95	ex 418	. . . 4 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ FinII → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)))
97	96	exlimiv 1962	. . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (𝐴 ∈ FinII → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)))
98	1, 97	sylbi 220	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinII → ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)))
99		relen 8985	. . . 4 ⊢ Rel ≈
100	99	brrelex2i 5708	. . 3 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ∈ V)
101		isfin2 10379	. . 3 ⊢ (𝐵 ∈ V → (𝐵 ∈ FinII ↔ ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)))
102	100, 101	syl 18	. 2 ⊢ (𝐴 ≈ 𝐵 → (𝐵 ∈ FinII ↔ ∀𝑥 ∈ 𝒫 𝒫 𝐵((𝑥 ≠ ∅ ∧ [⊊] Or 𝑥) → ∪ 𝑥 ∈ 𝑥)))
103	98, 102	sylibrd 262	1 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ FinII → 𝐵 ∈ FinII))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3896  ∅c0 4277  𝒫 cpw 4556  ∪ cuni 4866  ∪ ciun 4950   class class class wbr 5102   Or wor 5558  ^◡ccnv 5650  dom cdm 5651   “ cima 5654  –1-1→wf1 6540  –onto→wfo 6541  –1-1-onto→wf1o 6542   [⊊] crpss 7742   ≈ cen 8977  Fin^IIcfin2 10364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-rpss 7743  df-en 8981  df-fin2 10371

This theorem is used by: (None)