Theorem infmap2 10322

Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 10688 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 7451	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ↑m 𝐵) = (𝐴 ↑m ∅))
2		breq2 5110	. . . . 5 ⊢ (𝐵 = ∅ → (𝑥 ≈ 𝐵 ↔ 𝑥 ≈ ∅))
3	2	anbi2d 642	. . . 4 ⊢ (𝐵 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)))
4	3	abbidv 2829	. . 3 ⊢ (𝐵 = ∅ → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)})
5	1, 4	breq12d 5119	. 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 9007	. . . . . . . . . . 11 ⊢ Rel ≼
8	7	brrelex1i 5711	. . . . . . . . . 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 5712	. . . . . . . . . 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 9116	. . . . . . . . 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 9196	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑m 𝐵))
17	11, 9, 15, 16	syl3anc 1399	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑m 𝐵))
18		numdom 10144	. . . . . . . . . 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 10316	. . . . . . . . 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 9061	. . . . . . . 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 9198	. . . . . . 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 9006	. . . . . . 7 ⊢ Rel ≈
28	27	brrelex1i 5711	. . . . . 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 2901	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐴 ↑m 𝐵)} = (𝐴 ↑m 𝐵)
31		elmapi 8897	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ↑m 𝐵) → 𝑥:𝐵⟶𝐴)
32		fssxp 6759	. . . . . . . . 9 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ⊆ (𝐵 × 𝐴))
33		ffun 6734	. . . . . . . . . . 11 ⊢ (𝑥:𝐵⟶𝐴 → Fun 𝑥)
34		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
35	34	fundmen 9087	. . . . . . . . . . 11 ⊢ (Fun 𝑥 → dom 𝑥 ≈ 𝑥)
36		ensym 9058	. . . . . . . . . . 11 ⊢ (dom 𝑥 ≈ 𝑥 → 𝑥 ≈ dom 𝑥)
37	33, 35, 36	3syl 19	. . . . . . . . . 10 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ≈ dom 𝑥)
38		fdm 6741	. . . . . . . . . 10 ⊢ (𝑥:𝐵⟶𝐴 → dom 𝑥 = 𝐵)
39	37, 38	breqtrd 5134	. . . . . . . . 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 3994	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐴 ↑m 𝐵)} ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)}
43	30, 42	eqsstrri 3960	. . . . 5 ⊢ (𝐴 ↑m 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)}
44		ssdomg 9055	. . . . 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 9068	. . . 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 7476	. . . . . . 7 ⊢ (𝐴 ↑m 𝐵) ∈ V
49	48	mptex 7252	. . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∈ V
50	49	rnex 7946	. . . . 5 ⊢ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∈ V
51		ensym 9058	. . . . . . . . . . . 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 9011	. . . . . . . . . . 11 ⊢ (𝐵 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐵⟶1-1onto𝑥)
54	52, 53	sylib 221	. . . . . . . . . 10 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → ∃𝑓 𝑓:𝐵⟶1-1onto𝑥)
55		f1of 6846	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐵⟶1-1onto𝑥 → 𝑓:𝐵⟶𝑥)
56	55	adantl 487	. . . . . . . . . . . . . . 15 ⊢ (((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵⟶1-1onto𝑥) → 𝑓:𝐵⟶𝑥)
57		simplrl 789	. . . . . . . . . . . . . . 15 ⊢ (((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵⟶1-1onto𝑥) → 𝑥 ⊆ 𝐴)
58	56, 57	fssd 6749	. . . . . . . . . . . . . 14 ⊢ (((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵⟶1-1onto𝑥) → 𝑓:𝐵⟶𝐴)
59	11, 9	elmapd 8888	. . . . . . . . . . . . . . 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⟶1-1onto𝑥) → (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⟶1-1onto𝑥) → 𝑓 ∈ (𝐴 ↑m 𝐵))
62		f1ofo 6854	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐵⟶1-1onto𝑥 → 𝑓:𝐵⟶onto𝑥)
63		forn 6821	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐵⟶onto𝑥 → ran 𝑓 = 𝑥)
64	62, 63	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐵⟶1-1onto𝑥 → ran 𝑓 = 𝑥)
65	64	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵⟶1-1onto𝑥) → ran 𝑓 = 𝑥)
66	65	eqcomd 2769	. . . . . . . . . . . . 13 ⊢ (((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) ∧ 𝑓:𝐵⟶1-1onto𝑥) → 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⟶1-1onto𝑥) → (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⟶1-1onto𝑥 → (𝑓 ∈ (𝐴 ↑m 𝐵) ∧ 𝑥 = ran 𝑓)))
69	68	eximdv 1952	. . . . . . . . . 10 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → (∃𝑓 𝑓:𝐵⟶1-1onto𝑥 → ∃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 3091	. . . . . . . . 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 3993	. . . . . 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 2763	. . . . . . 7 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) = (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓)
76	75	rnmpt 5946	. . . . . 6 ⊢ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) = {𝑥 ∣ ∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓}
77	74, 76	sseqtrrdi 3954	. . . . 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 9055	. . . . 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 3458	. . . . . . . . 9 ⊢ 𝑓 ∈ V
81	80	rnex 7946	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
82	81	rgenw 3084	. . . . . . 7 ⊢ ∀𝑓 ∈ (𝐴 ↑m 𝐵)ran 𝑓 ∈ V
83	75	fnmpt 6701	. . . . . . 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 6824	. . . . . 6 ⊢ ((𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) Fn (𝐴 ↑m 𝐵) ↔ (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓):(𝐴 ↑m 𝐵)⟶ontoran (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)⟶ontoran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓))
87		fodomnum 10163	. . . . 5 ⊢ ((𝐴 ↑m 𝐵) ∈ dom card → ((𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓):(𝐴 ↑m 𝐵)⟶ontoran (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 9062	. . . 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 9144	. . 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 5712	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
94	93	3ad2ant1 1151	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → 𝐴 ∈ V)
95		map0e 8938	. . . 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 8514	. . . . 5 ⊢ 1o ∈ V
98	97	enref 9040	. . . 4 ⊢ 1o ≈ 1o
99		df-sn 4589	. . . . 5 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
100		df1o2 8511	. . . . 5 ⊢ 1o = {∅}
101		en0 9073	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
102	101	anbi2i 635	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = ∅))
103		0ss 4353	. . . . . . . . 9 ⊢ ∅ ⊆ 𝐴
104		sseq1 3938	. . . . . . . . 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 2830	. . . . 5 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)} = {𝑥 ∣ 𝑥 = ∅}
109	99, 100, 108	3eqtr4ri 2797	. . . 4 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)} = 1o
110	98, 109	breqtrri 5135	. . 3 ⊢ 1o ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)}
111	96, 110	eqbrtrdi 5147	. 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → (𝐴 ↑m ∅) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)})
112	5, 92, 111	pm2.61ne 3044	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 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  {csn 4588   class class class wbr 5106   ↦ cmpt 5189   × cxp 5653  dom cdm 5655  ran crn 5656  Fun wfun 6552   Fn wfn 6553  ⟶wf 6554  ⟶ontowfo 6556  ⟶1-1ontowf1o 6557  (class class class)co 7443  ωcom 7901  1_oc1o 8497   ↑_m cmap 8875   ≈ cen 8998   ≼ cdom 8999  cardccrd 10043

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-er 8745  df-map 8877  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-oi 9531  df-card 10047  df-acn 10050

This theorem is used by:  infmap  10688