Theorem infmap2 10267

Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 10633 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
infmap2	⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem infmap2

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7416	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ↑m 𝐵) = (𝐴 ↑m ∅))
2		breq2 5106	. . . . 5 ⊢ (𝐵 = ∅ → (𝑥 ≈ 𝐵 ↔ 𝑥 ≈ ∅))
3	2	anbi2d 642	. . . 4 ⊢ (𝐵 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)))
4	3	abbidv 2826	. . 3 ⊢ (𝐵 = ∅ → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)})
5	1, 4	breq12d 5115	. 2 ⊢ (𝐵 = ∅ → ((𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ↔ (𝐴 ↑m ∅) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)}))
6		simpl2 1211	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≼ 𝐴)
7		reldom 8957	. . . . . . . . . . 11 ⊢ Rel ≼
8	7	brrelex1i 5703	. . . . . . . . . 10 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V)
9	6, 8	syl 18	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ∈ V)
10	7	brrelex2i 5704	. . . . . . . . . 10 ⊢ (𝐵 ≼ 𝐴 → 𝐴 ∈ V)
11	6, 10	syl 18	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V)
12		xpcomeng 9066	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
13	9, 11, 12	syl2anc 596	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
14		simpl3 1212	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) ∈ dom card)
15		simpr 490	. . . . . . . . . . 11 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
16		mapdom3 9146	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑m 𝐵))
17	11, 9, 15, 16	syl3anc 1398	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑m 𝐵))
18		numdom 10089	. . . . . . . . . 10 ⊢ (((𝐴 ↑m 𝐵) ∈ dom card ∧ 𝐴 ≼ (𝐴 ↑m 𝐵)) → 𝐴 ∈ dom card)
19	14, 17, 18	syl2anc 596	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ dom card)
20		simpl1 1210	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ω ≼ 𝐴)
21		infxpabs 10261	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
22	19, 20, 15, 6, 21	syl22anc 852	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≈ 𝐴)
23		entr 9011	. . . . . . . 8 ⊢ (((𝐵 × 𝐴) ≈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ≈ 𝐴) → (𝐵 × 𝐴) ≈ 𝐴)
24	13, 22, 23	syl2anc 596	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ 𝐴)
25		ssenen 9148	. . . . . . 7 ⊢ ((𝐵 × 𝐴) ≈ 𝐴 → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
26	24, 25	syl 18	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
27		relen 8956	. . . . . . 7 ⊢ Rel ≈
28	27	brrelex1i 5703	. . . . . 6 ⊢ ({𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ∈ V)
29	26, 28	syl 18	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ∈ V)
30		abid2 2897	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐴 ↑m 𝐵)} = (𝐴 ↑m 𝐵)
31		elmapi 8847	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ↑m 𝐵) → 𝑥:𝐵⟶𝐴)
32		fssxp 6725	. . . . . . . . 9 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ⊆ (𝐵 × 𝐴))
33		ffun 6700	. . . . . . . . . . 11 ⊢ (𝑥:𝐵⟶𝐴 → Fun 𝑥)
34		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
35	34	fundmen 9037	. . . . . . . . . . 11 ⊢ (Fun 𝑥 → dom 𝑥 ≈ 𝑥)
36		ensym 9008	. . . . . . . . . . 11 ⊢ (dom 𝑥 ≈ 𝑥 → 𝑥 ≈ dom 𝑥)
37	33, 35, 36	3syl 19	. . . . . . . . . 10 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ≈ dom 𝑥)
38		fdm 6707	. . . . . . . . . 10 ⊢ (𝑥:𝐵⟶𝐴 → dom 𝑥 = 𝐵)
39	37, 38	breqtrd 5130	. . . . . . . . 9 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ≈ 𝐵)
40	32, 39	jca 521	. . . . . . . 8 ⊢ (𝑥:𝐵⟶𝐴 → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵))
41	31, 40	syl 18	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↑m 𝐵) → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵))
42	41	ss2abi 4013	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐴 ↑m 𝐵)} ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)}
43	30, 42	eqsstrri 3977	. . . . 5 ⊢ (𝐴 ↑m 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)}
44		ssdomg 9005	. . . . 5 ⊢ ({𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ∈ V → ((𝐴 ↑m 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} → (𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)}))
45	29, 43, 44	mpisyl 22	. . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)})
46		domentr 9018	. . . 4 ⊢ (((𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ∧ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)}) → (𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
47	45, 26, 46	syl2anc 596	. . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
48		ovex 7441	. . . . . . 7 ⊢ (𝐴 ↑m 𝐵) ∈ V
49	48	mptex 7217	. . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∈ V
50	49	rnex 7905	. . . . 5 ⊢ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∈ V
51		ensym 9008	. . . . . . . . . . . 12 ⊢ (𝑥 ≈ 𝐵 → 𝐵 ≈ 𝑥)
52	51	ad2antll 742	. . . . . . . . . . 11 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → 𝐵 ≈ 𝑥)
53		bren 8961	. . . . . . . . . . 11 ⊢ (𝐵 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝑥)
54	52, 53	sylib 221	. . . . . . . . . 10 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → ∃𝑓 𝑓:𝐵–1-1-onto→𝑥)
55		f1of 6812	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐵–1-1-onto→𝑥 → 𝑓:𝐵⟶𝑥)
56	55	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → 𝑓:𝐵⟶𝑥)
57		simplrl 789	. . . . . . . . . . . . . . 15 ⊢ (((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → 𝑥 ⊆ 𝐴)
58	56, 57	fssd 6715	. . . . . . . . . . . . . 14 ⊢ (((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → 𝑓:𝐵⟶𝐴)
59	11, 9	elmapd 8838	. . . . . . . . . . . . . . 15 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴 ↑m 𝐵) ↔ 𝑓:𝐵⟶𝐴))
60	59	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → (𝑓 ∈ (𝐴 ↑m 𝐵) ↔ 𝑓:𝐵⟶𝐴))
61	58, 60	mpbird 260	. . . . . . . . . . . . 13 ⊢ (((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → 𝑓 ∈ (𝐴 ↑m 𝐵))
62		f1ofo 6820	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐵–1-1-onto→𝑥 → 𝑓:𝐵–onto→𝑥)
63		forn 6787	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐵–onto→𝑥 → ran 𝑓 = 𝑥)
64	62, 63	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝑥 → ran 𝑓 = 𝑥)
65	64	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → ran 𝑓 = 𝑥)
66	65	eqcomd 2766	. . . . . . . . . . . . 13 ⊢ (((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → 𝑥 = ran 𝑓)
67	61, 66	jca 521	. . . . . . . . . . . 12 ⊢ (((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵–1-1-onto→𝑥) → (𝑓 ∈ (𝐴 ↑m 𝐵) ∧ 𝑥 = ran 𝑓))
68	67	ex 418	. . . . . . . . . . 11 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → (𝑓:𝐵–1-1-onto→𝑥 → (𝑓 ∈ (𝐴 ↑m 𝐵) ∧ 𝑥 = ran 𝑓)))
69	68	eximdv 1950	. . . . . . . . . 10 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → (∃𝑓 𝑓:𝐵–1-1-onto→𝑥 → ∃𝑓(𝑓 ∈ (𝐴 ↑m 𝐵) ∧ 𝑥 = ran 𝑓)))
70	54, 69	mpd 16	. . . . . . . . 9 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → ∃𝑓(𝑓 ∈ (𝐴 ↑m 𝐵) ∧ 𝑥 = ran 𝑓))
71		df-rex 3087	. . . . . . . . 9 ⊢ (∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓 ↔ ∃𝑓(𝑓 ∈ (𝐴 ↑m 𝐵) ∧ 𝑥 = ran 𝑓))
72	70, 71	sylibr 237	. . . . . . . 8 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → ∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓)
73	72	ex 418	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵) → ∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓))
74	73	ss2abdv 4012	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ⊆ {𝑥 ∣ ∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓})
75		eqid 2760	. . . . . . 7 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) = (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓)
76	75	rnmpt 5935	. . . . . 6 ⊢ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) = {𝑥 ∣ ∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓}
77	74, 76	sseqtrrdi 3971	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ⊆ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓))
78		ssdomg 9005	. . . . 5 ⊢ (ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∈ V → ({𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ⊆ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓)))
79	50, 77, 78	mpsyl 69	. . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓))
80		vex 3454	. . . . . . . . 9 ⊢ 𝑓 ∈ V
81	80	rnex 7905	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
82	81	rgenw 3080	. . . . . . 7 ⊢ ∀𝑓 ∈ (𝐴 ↑m 𝐵)ran 𝑓 ∈ V
83	75	fnmpt 6667	. . . . . . 7 ⊢ (∀𝑓 ∈ (𝐴 ↑m 𝐵)ran 𝑓 ∈ V → (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) Fn (𝐴 ↑m 𝐵))
84	82, 83	mp1i 14	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) Fn (𝐴 ↑m 𝐵))
85		dffn4 6790	. . . . . 6 ⊢ ((𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) Fn (𝐴 ↑m 𝐵) ↔ (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓):(𝐴 ↑m 𝐵)–onto→ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓))
86	84, 85	sylib 221	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓):(𝐴 ↑m 𝐵)–onto→ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓))
87		fodomnum 10108	. . . . 5 ⊢ ((𝐴 ↑m 𝐵) ∈ dom card → ((𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓):(𝐴 ↑m 𝐵)–onto→ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) → ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ≼ (𝐴 ↑m 𝐵)))
88	14, 86, 87	sylc 66	. . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ≼ (𝐴 ↑m 𝐵))
89		domtr 9012	. . . 4 ⊢ (({𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∧ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ≼ (𝐴 ↑m 𝐵)) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ (𝐴 ↑m 𝐵))
90	79, 88, 89	syl2anc 596	. . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ (𝐴 ↑m 𝐵))
91		sbth 9094	. . 3 ⊢ (((𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ∧ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ (𝐴 ↑m 𝐵)) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
92	47, 90, 91	syl2anc 596	. 2 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
93	7	brrelex2i 5704	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
94	93	3ad2ant1 1151	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → 𝐴 ∈ V)
95		map0e 8888	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ↑m ∅) = 1o)
96	94, 95	syl 18	. . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → (𝐴 ↑m ∅) = 1o)
97		1oex 8464	. . . . 5 ⊢ 1o ∈ V
98	97	enref 8990	. . . 4 ⊢ 1o ≈ 1o
99		df-sn 4584	. . . . 5 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
100		df1o2 8461	. . . . 5 ⊢ 1o = {∅}
101		en0 9023	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
102	101	anbi2i 635	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = ∅))
103		0ss 4349	. . . . . . . . 9 ⊢ ∅ ⊆ 𝐴
104		sseq1 3955	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
105	103, 104	mpbiri 261	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝑥 ⊆ 𝐴)
106	105	pm4.71ri 570	. . . . . . 7 ⊢ (𝑥 = ∅ ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = ∅))
107	102, 106	bitr4i 281	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ 𝑥 = ∅)
108	107	abbii 2827	. . . . 5 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)} = {𝑥 ∣ 𝑥 = ∅}
109	99, 100, 108	3eqtr4ri 2794	. . . 4 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)} = 1o
110	98, 109	breqtrri 5131	. . 3 ⊢ 1o ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)}
111	96, 110	eqbrtrdi 5143	. 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → (𝐴 ↑m ∅) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)})
112	5, 92, 111	pm2.61ne 3040	1 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3898  ∅c0 4278  {csn 4583   class class class wbr 5102   ↦ cmpt 5185   × cxp 5645  dom cdm 5647  ran crn 5648  Fun wfun 6521   Fn wfn 6522  ⟶wf 6523  –onto→wfo 6525  –1-1-onto→wf1o 6526  (class class class)co 7408  ωcom 7860  1_oc1o 8447   ↑_m cmap 8825   ≈ cen 8948   ≼ cdom 8949  cardccrd 9988

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-er 8695  df-map 8827  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-oi 9482  df-card 9992  df-acn 9995

This theorem is used by:  infmap  10633