Theorem infmap2 9040

Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 9398 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 6658	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ↑𝑚 𝐵) = (𝐴 ↑𝑚 ∅))
2		breq2 4657	. . . . 5 ⊢ (𝐵 = ∅ → (𝑥 ≈ 𝐵 ↔ 𝑥 ≈ ∅))
3	2	anbi2d 740	. . . 4 ⊢ (𝐵 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)))
4	3	abbidv 2741	. . 3 ⊢ (𝐵 = ∅ → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)})
5	1, 4	breq12d 4666	. 2 ⊢ (𝐵 = ∅ → ((𝐴 ↑𝑚 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ↔ (𝐴 ↑𝑚 ∅) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)}))
6		simpl2 1065	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≼ 𝐴)
7		reldom 7961	. . . . . . . . . . 11 ⊢ Rel ≼
8	7	brrelexi 5158	. . . . . . . . . 10 ⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V)
9	6, 8	syl 17	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ∈ V)
10	7	brrelex2i 5159	. . . . . . . . . 10 ⊢ (𝐵 ≼ 𝐴 → 𝐴 ∈ V)
11	6, 10	syl 17	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V)
12		xpcomeng 8052	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
13	9, 11, 12	syl2anc 693	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
14		simpl3 1066	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑𝑚 𝐵) ∈ dom card)
15		simpr 477	. . . . . . . . . . 11 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
16		mapdom3 8132	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑𝑚 𝐵))
17	11, 9, 15, 16	syl3anc 1326	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑𝑚 𝐵))
18		numdom 8861	. . . . . . . . . 10 ⊢ (((𝐴 ↑𝑚 𝐵) ∈ dom card ∧ 𝐴 ≼ (𝐴 ↑𝑚 𝐵)) → 𝐴 ∈ dom card)
19	14, 17, 18	syl2anc 693	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ dom card)
20		simpl1 1064	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ω ≼ 𝐴)
21		infxpabs 9034	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
22	19, 20, 15, 6, 21	syl22anc 1327	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≈ 𝐴)
23		entr 8008	. . . . . . . 8 ⊢ (((𝐵 × 𝐴) ≈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ≈ 𝐴) → (𝐵 × 𝐴) ≈ 𝐴)
24	13, 22, 23	syl2anc 693	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ 𝐴)
25		ssenen 8134	. . . . . . 7 ⊢ ((𝐵 × 𝐴) ≈ 𝐴 → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
26	24, 25	syl 17	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
27		relen 7960	. . . . . . 7 ⊢ Rel ≈
28	27	brrelexi 5158	. . . . . 6 ⊢ ({𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ∈ V)
29	26, 28	syl 17	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)} ∈ V)
30		abid2 2745	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐴 ↑𝑚 𝐵)} = (𝐴 ↑𝑚 𝐵)
31		elmapi 7879	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 𝐵) → 𝑥:𝐵⟶𝐴)
32		fssxp 6060	. . . . . . . . 9 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ⊆ (𝐵 × 𝐴))
33		ffun 6048	. . . . . . . . . . 11 ⊢ (𝑥:𝐵⟶𝐴 → Fun 𝑥)
34		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
35	34	fundmen 8030	. . . . . . . . . . 11 ⊢ (Fun 𝑥 → dom 𝑥 ≈ 𝑥)
36		ensym 8005	. . . . . . . . . . 11 ⊢ (dom 𝑥 ≈ 𝑥 → 𝑥 ≈ dom 𝑥)
37	33, 35, 36	3syl 18	. . . . . . . . . 10 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ≈ dom 𝑥)
38		fdm 6051	. . . . . . . . . 10 ⊢ (𝑥:𝐵⟶𝐴 → dom 𝑥 = 𝐵)
39	37, 38	breqtrd 4679	. . . . . . . . 9 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ≈ 𝐵)
40	32, 39	jca 554	. . . . . . . 8 ⊢ (𝑥:𝐵⟶𝐴 → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵))
41	31, 40	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 𝐵) → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵))
42	41	ss2abi 3674	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐴 ↑𝑚 𝐵)} ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)}
43	30, 42	eqsstr3i 3636	. . . . 5 ⊢ (𝐴 ↑𝑚 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)}
44		ssdomg 8001	. . . . 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 21	. . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑𝑚 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)})
46		domentr 8015	. . . 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 693	. . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑𝑚 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
48		ovex 6678	. . . . . . 7 ⊢ (𝐴 ↑𝑚 𝐵) ∈ V
49	48	mptex 6486	. . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐵) ↦ ran 𝑓) ∈ V
50	49	rnex 7100	. . . . 5 ⊢ ran (𝑓 ∈ (𝐴 ↑𝑚 𝐵) ↦ ran 𝑓) ∈ V
51		ensym 8005	. . . . . . . . . . . 12 ⊢ (𝑥 ≈ 𝐵 → 𝐵 ≈ 𝑥)
52	51	ad2antll 765	. . . . . . . . . . 11 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → 𝐵 ≈ 𝑥)
53		bren 7964	. . . . . . . . . . 11 ⊢ (𝐵 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝑥)
54	52, 53	sylib 208	. . . . . . . . . 10 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → ∃𝑓 𝑓:𝐵–1-1-onto→𝑥)
55		f1of 6137	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐵–1-1-onto→𝑥 → 𝑓:𝐵⟶𝑥)
56	55	adantl 482	. . . . . . . . . . . . . . 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 800	. . . . . . . . . . . . . . 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 6057	. . . . . . . . . . . . . 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 7871	. . . . . . . . . . . . . . 15 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝑓:𝐵⟶𝐴))
60	59	ad2antrr 762	. . . . . . . . . . . . . 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 247	. . . . . . . . . . . . 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 6144	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐵–1-1-onto→𝑥 → 𝑓:𝐵–onto→𝑥)
63		forn 6118	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐵–onto→𝑥 → ran 𝑓 = 𝑥)
64	62, 63	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵–1-1-onto→𝑥 → ran 𝑓 = 𝑥)
65	64	adantl 482	. . . . . . . . . . . . . 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 2628	. . . . . . . . . . . . 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 554	. . . . . . . . . . . 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 450	. . . . . . . . . . 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 1846	. . . . . . . . . 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 15	. . . . . . . . 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 2918	. . . . . . . . 9 ⊢ (∃𝑓 ∈ (𝐴 ↑𝑚 𝐵)𝑥 = ran 𝑓 ↔ ∃𝑓(𝑓 ∈ (𝐴 ↑𝑚 𝐵) ∧ 𝑥 = ran 𝑓))
72	70, 71	sylibr 224	. . . . . . . 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 450	. . . . . . 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 3675	. . . . . 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 2622	. . . . . . 7 ⊢ (𝑓 ∈ (𝐴 ↑𝑚 𝐵) ↦ ran 𝑓) = (𝑓 ∈ (𝐴 ↑𝑚 𝐵) ↦ ran 𝑓)
76	75	rnmpt 5371	. . . . . 6 ⊢ ran (𝑓 ∈ (𝐴 ↑𝑚 𝐵) ↦ ran 𝑓) = {𝑥 ∣ ∃𝑓 ∈ (𝐴 ↑𝑚 𝐵)𝑥 = ran 𝑓}
77	74, 76	syl6sseqr 3652	. . . . 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 8001	. . . . 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 68	. . . 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 3203	. . . . . . . . 9 ⊢ 𝑓 ∈ V
81	80	rnex 7100	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
82	81	rgenw 2924	. . . . . . 7 ⊢ ∀𝑓 ∈ (𝐴 ↑𝑚 𝐵)ran 𝑓 ∈ V
83	75	fnmpt 6020	. . . . . . 7 ⊢ (∀𝑓 ∈ (𝐴 ↑𝑚 𝐵)ran 𝑓 ∈ V → (𝑓 ∈ (𝐴 ↑𝑚 𝐵) ↦ ran 𝑓) Fn (𝐴 ↑𝑚 𝐵))
84	82, 83	mp1i 13	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴 ↑𝑚 𝐵) ↦ ran 𝑓) Fn (𝐴 ↑𝑚 𝐵))
85		dffn4 6121	. . . . . 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 208	. . . . 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 8880	. . . . 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 65	. . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ran (𝑓 ∈ (𝐴 ↑𝑚 𝐵) ↦ ran 𝑓) ≼ (𝐴 ↑𝑚 𝐵))
89		domtr 8009	. . . 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 693	. . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ (𝐴 ↑𝑚 𝐵))
91		sbth 8080	. . 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 693	. 2 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑𝑚 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
93	7	brrelex2i 5159	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
94	93	3ad2ant1 1082	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) → 𝐴 ∈ V)
95		map0e 7895	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ↑𝑚 ∅) = 1𝑜)
96	94, 95	syl 17	. . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) → (𝐴 ↑𝑚 ∅) = 1𝑜)
97		1onn 7719	. . . . . 6 ⊢ 1𝑜 ∈ ω
98	97	elexi 3213	. . . . 5 ⊢ 1𝑜 ∈ V
99	98	enref 7988	. . . 4 ⊢ 1𝑜 ≈ 1𝑜
100		df-sn 4178	. . . . 5 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
101		df1o2 7572	. . . . 5 ⊢ 1𝑜 = {∅}
102		en0 8019	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
103	102	anbi2i 730	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = ∅))
104		0ss 3972	. . . . . . . . 9 ⊢ ∅ ⊆ 𝐴
105		sseq1 3626	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
106	104, 105	mpbiri 248	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝑥 ⊆ 𝐴)
107	106	pm4.71ri 665	. . . . . . 7 ⊢ (𝑥 = ∅ ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = ∅))
108	103, 107	bitr4i 267	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ 𝑥 = ∅)
109	108	abbii 2739	. . . . 5 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)} = {𝑥 ∣ 𝑥 = ∅}
110	100, 101, 109	3eqtr4ri 2655	. . . 4 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)} = 1𝑜
111	99, 110	breqtrri 4680	. . 3 ⊢ 1𝑜 ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)}
112	96, 111	syl6eqbr 4692	. 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) → (𝐴 ↑𝑚 ∅) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)})
113	5, 92, 112	pm2.61ne 2879	1 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑𝑚 𝐵) ∈ dom card) → (𝐴 ↑𝑚 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  {csn 4177   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  ran crn 5115  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  (class class class)co 6650  ωcom 7065  1_𝑜c1o 7553   ↑_𝑚 cmap 7857   ≈ cen 7952   ≼ cdom 7953  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-acn 8768

This theorem is referenced by:  infmap  9398