Theorem infmap2 10286

Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 10645 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 7456	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ↑m 𝐵) = (𝐴 ↑m ∅))
2		breq2 5170	. . . . 5 ⊢ (𝐵 = ∅ → (𝑥 ≈ 𝐵 ↔ 𝑥 ≈ ∅))
3	2	anbi2d 629	. . . 4 ⊢ (𝐵 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)))
4	3	abbidv 2811	. . 3 ⊢ (𝐵 = ∅ → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)})
5	1, 4	breq12d 5179	. 2 ⊢ (𝐵 = ∅ → ((𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ↔ (𝐴 ↑m ∅) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)}))
6		simpl2 1192	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≼ 𝐴)
7		reldom 9009	. . . . . . . . . . 11 ⊢ Rel ≼
8	7	brrelex1i 5756	. . . . . . . . . 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 5757	. . . . . . . . . 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 9130	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
13	9, 11, 12	syl2anc 583	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
14		simpl3 1193	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) ∈ dom card)
15		simpr 484	. . . . . . . . . . 11 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
16		mapdom3 9215	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑m 𝐵))
17	11, 9, 15, 16	syl3anc 1371	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑m 𝐵))
18		numdom 10107	. . . . . . . . . 10 ⊢ (((𝐴 ↑m 𝐵) ∈ dom card ∧ 𝐴 ≼ (𝐴 ↑m 𝐵)) → 𝐴 ∈ dom card)
19	14, 17, 18	syl2anc 583	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ dom card)
20		simpl1 1191	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ω ≼ 𝐴)
21		infxpabs 10280	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
22	19, 20, 15, 6, 21	syl22anc 838	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≈ 𝐴)
23		entr 9066	. . . . . . . 8 ⊢ (((𝐵 × 𝐴) ≈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ≈ 𝐴) → (𝐵 × 𝐴) ≈ 𝐴)
24	13, 22, 23	syl2anc 583	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ 𝐴)
25		ssenen 9217	. . . . . . 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 9008	. . . . . . 7 ⊢ Rel ≈
28	27	brrelex1i 5756	. . . . . 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 2882	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐴 ↑m 𝐵)} = (𝐴 ↑m 𝐵)
31		elmapi 8907	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ↑m 𝐵) → 𝑥:𝐵⟶𝐴)
32		fssxp 6775	. . . . . . . . 9 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ⊆ (𝐵 × 𝐴))
33		ffun 6750	. . . . . . . . . . 11 ⊢ (𝑥:𝐵⟶𝐴 → Fun 𝑥)
34		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
35	34	fundmen 9096	. . . . . . . . . . 11 ⊢ (Fun 𝑥 → dom 𝑥 ≈ 𝑥)
36		ensym 9063	. . . . . . . . . . 11 ⊢ (dom 𝑥 ≈ 𝑥 → 𝑥 ≈ dom 𝑥)
37	33, 35, 36	3syl 18	. . . . . . . . . 10 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ≈ dom 𝑥)
38		fdm 6756	. . . . . . . . . 10 ⊢ (𝑥:𝐵⟶𝐴 → dom 𝑥 = 𝐵)
39	37, 38	breqtrd 5192	. . . . . . . . 9 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ≈ 𝐵)
40	32, 39	jca 511	. . . . . . . 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 4090	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐴 ↑m 𝐵)} ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)}
43	30, 42	eqsstrri 4044	. . . . 5 ⊢ (𝐴 ↑m 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)}
44		ssdomg 9060	. . . . 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 9073	. . . 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 583	. . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
48		ovex 7481	. . . . . . 7 ⊢ (𝐴 ↑m 𝐵) ∈ V
49	48	mptex 7260	. . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∈ V
50	49	rnex 7950	. . . . 5 ⊢ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∈ V
51		ensym 9063	. . . . . . . . . . . 12 ⊢ (𝑥 ≈ 𝐵 → 𝐵 ≈ 𝑥)
52	51	ad2antll 728	. . . . . . . . . . 11 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → 𝐵 ≈ 𝑥)
53		bren 9013	. . . . . . . . . . 11 ⊢ (𝐵 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝑥)
54	52, 53	sylib 218	. . . . . . . . . 10 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → ∃𝑓 𝑓:𝐵–1-1-onto→𝑥)
55		f1of 6862	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐵–1-1-onto→𝑥 → 𝑓:𝐵⟶𝑥)
56	55	adantl 481	. . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . 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 6764	. . . . . . . . . . . . . 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 8898	. . . . . . . . . . . . . . 15 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴 ↑m 𝐵) ↔ 𝑓:𝐵⟶𝐴))
60	59	ad2antrr 725	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 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 6869	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐵–1-1-onto→𝑥 → 𝑓:𝐵–onto→𝑥)
63		forn 6837	. . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 2746	. . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 1916	. . . . . . . . . 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 3077	. . . . . . . . 9 ⊢ (∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓 ↔ ∃𝑓(𝑓 ∈ (𝐴 ↑m 𝐵) ∧ 𝑥 = ran 𝑓))
72	70, 71	sylibr 234	. . . . . . . 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 412	. . . . . . 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 4089	. . . . . 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 2740	. . . . . . 7 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) = (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓)
76	75	rnmpt 5980	. . . . . 6 ⊢ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) = {𝑥 ∣ ∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓}
77	74, 76	sseqtrrdi 4060	. . . . 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 9060	. . . . 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 3492	. . . . . . . . 9 ⊢ 𝑓 ∈ V
81	80	rnex 7950	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
82	81	rgenw 3071	. . . . . . 7 ⊢ ∀𝑓 ∈ (𝐴 ↑m 𝐵)ran 𝑓 ∈ V
83	75	fnmpt 6720	. . . . . . 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 6840	. . . . . 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 218	. . . . 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 10126	. . . . 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 9067	. . . 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 583	. . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ (𝐴 ↑m 𝐵))
91		sbth 9159	. . 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 583	. 2 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
93	7	brrelex2i 5757	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
94	93	3ad2ant1 1133	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → 𝐴 ∈ V)
95		map0e 8940	. . . 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		1oex 8532	. . . . 5 ⊢ 1o ∈ V
98	97	enref 9045	. . . 4 ⊢ 1o ≈ 1o
99		df-sn 4649	. . . . 5 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
100		df1o2 8529	. . . . 5 ⊢ 1o = {∅}
101		en0 9078	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
102	101	anbi2i 622	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = ∅))
103		0ss 4423	. . . . . . . . 9 ⊢ ∅ ⊆ 𝐴
104		sseq1 4034	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
105	103, 104	mpbiri 258	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝑥 ⊆ 𝐴)
106	105	pm4.71ri 560	. . . . . . 7 ⊢ (𝑥 = ∅ ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = ∅))
107	102, 106	bitr4i 278	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ 𝑥 = ∅)
108	107	abbii 2812	. . . . 5 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)} = {𝑥 ∣ 𝑥 = ∅}
109	99, 100, 108	3eqtr4ri 2779	. . . 4 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)} = 1o
110	98, 109	breqtrri 5193	. . 3 ⊢ 1o ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)}
111	96, 110	eqbrtrdi 5205	. 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → (𝐴 ↑m ∅) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)})
112	5, 92, 111	pm2.61ne 3033	1 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  {csn 4648   class class class wbr 5166   ↦ cmpt 5249   × cxp 5698  dom cdm 5700  ran crn 5701  Fun wfun 6567   Fn wfn 6568  ⟶wf 6569  –onto→wfo 6571  –1-1-onto→wf1o 6572  (class class class)co 7448  ωcom 7903  1_oc1o 8515   ↑_m cmap 8884   ≈ cen 9000   ≼ cdom 9001  cardccrd 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-oi 9579  df-card 10008  df-acn 10011

This theorem is referenced by:  infmap  10645