Theorem infmap2 10163

Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 10521 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 7370	. . 3 ⊢ (𝐵 = ∅ → (𝐴 ↑m 𝐵) = (𝐴 ↑m ∅))
2		breq2 5114	. . . . 5 ⊢ (𝐵 = ∅ → (𝑥 ≈ 𝐵 ↔ 𝑥 ≈ ∅))
3	2	anbi2d 629	. . . 4 ⊢ (𝐵 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)))
4	3	abbidv 2800	. . 3 ⊢ (𝐵 = ∅ → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)})
5	1, 4	breq12d 5123	. 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 8896	. . . . . . . . . . 11 ⊢ Rel ≼
8	7	brrelex1i 5693	. . . . . . . . . 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 5694	. . . . . . . . . 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 9015	. . . . . . . . 9 ⊢ ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
13	9, 11, 12	syl2anc 584	. . . . . . . 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 485	. . . . . . . . . . 11 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
16		mapdom3 9100	. . . . . . . . . . 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 9983	. . . . . . . . . 10 ⊢ (((𝐴 ↑m 𝐵) ∈ dom card ∧ 𝐴 ≼ (𝐴 ↑m 𝐵)) → 𝐴 ∈ dom card)
19	14, 17, 18	syl2anc 584	. . . . . . . . 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 10157	. . . . . . . . 9 ⊢ (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵 ≼ 𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
22	19, 20, 15, 6, 21	syl22anc 837	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≈ 𝐴)
23		entr 8953	. . . . . . . 8 ⊢ (((𝐵 × 𝐴) ≈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ≈ 𝐴) → (𝐵 × 𝐴) ≈ 𝐴)
24	13, 22, 23	syl2anc 584	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ 𝐴)
25		ssenen 9102	. . . . . . 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 8895	. . . . . . 7 ⊢ Rel ≈
28	27	brrelex1i 5693	. . . . . 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 2870	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐴 ↑m 𝐵)} = (𝐴 ↑m 𝐵)
31		elmapi 8794	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ↑m 𝐵) → 𝑥:𝐵⟶𝐴)
32		fssxp 6701	. . . . . . . . 9 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ⊆ (𝐵 × 𝐴))
33		ffun 6676	. . . . . . . . . . 11 ⊢ (𝑥:𝐵⟶𝐴 → Fun 𝑥)
34		vex 3450	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
35	34	fundmen 8982	. . . . . . . . . . 11 ⊢ (Fun 𝑥 → dom 𝑥 ≈ 𝑥)
36		ensym 8950	. . . . . . . . . . 11 ⊢ (dom 𝑥 ≈ 𝑥 → 𝑥 ≈ dom 𝑥)
37	33, 35, 36	3syl 18	. . . . . . . . . 10 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ≈ dom 𝑥)
38		fdm 6682	. . . . . . . . . 10 ⊢ (𝑥:𝐵⟶𝐴 → dom 𝑥 = 𝐵)
39	37, 38	breqtrd 5136	. . . . . . . . 9 ⊢ (𝑥:𝐵⟶𝐴 → 𝑥 ≈ 𝐵)
40	32, 39	jca 512	. . . . . . . 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 4028	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 ∈ (𝐴 ↑m 𝐵)} ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)}
43	30, 42	eqsstrri 3982	. . . . 5 ⊢ (𝐴 ↑m 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥 ≈ 𝐵)}
44		ssdomg 8947	. . . . 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 8960	. . . 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 584	. . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
48		ovex 7395	. . . . . . 7 ⊢ (𝐴 ↑m 𝐵) ∈ V
49	48	mptex 7178	. . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∈ V
50	49	rnex 7854	. . . . 5 ⊢ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) ∈ V
51		ensym 8950	. . . . . . . . . . . 12 ⊢ (𝑥 ≈ 𝐵 → 𝐵 ≈ 𝑥)
52	51	ad2antll 727	. . . . . . . . . . 11 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → 𝐵 ≈ 𝑥)
53		bren 8900	. . . . . . . . . . 11 ⊢ (𝐵 ≈ 𝑥 ↔ ∃𝑓 𝑓:𝐵–1-1-onto→𝑥)
54	52, 53	sylib 217	. . . . . . . . . 10 ⊢ ((((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)) → ∃𝑓 𝑓:𝐵–1-1-onto→𝑥)
55		f1of 6789	. . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . 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 6691	. . . . . . . . . . . . . 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 8786	. . . . . . . . . . . . . . 15 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴 ↑m 𝐵) ↔ 𝑓:𝐵⟶𝐴))
60	59	ad2antrr 724	. . . . . . . . . . . . . 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 256	. . . . . . . . . . . . 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 6796	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:𝐵–1-1-onto→𝑥 → 𝑓:𝐵–onto→𝑥)
63		forn 6764	. . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . 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 512	. . . . . . . . . . . 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 413	. . . . . . . . . . 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 1920	. . . . . . . . . 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 3070	. . . . . . . . 9 ⊢ (∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓 ↔ ∃𝑓(𝑓 ∈ (𝐴 ↑m 𝐵) ∧ 𝑥 = ran 𝑓))
72	70, 71	sylibr 233	. . . . . . . 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 413	. . . . . . 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 4025	. . . . . 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 2731	. . . . . . 7 ⊢ (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) = (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓)
76	75	rnmpt 5915	. . . . . 6 ⊢ ran (𝑓 ∈ (𝐴 ↑m 𝐵) ↦ ran 𝑓) = {𝑥 ∣ ∃𝑓 ∈ (𝐴 ↑m 𝐵)𝑥 = ran 𝑓}
77	74, 76	sseqtrrdi 3998	. . . . 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 8947	. . . . 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 3450	. . . . . . . . 9 ⊢ 𝑓 ∈ V
81	80	rnex 7854	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
82	81	rgenw 3064	. . . . . . 7 ⊢ ∀𝑓 ∈ (𝐴 ↑m 𝐵)ran 𝑓 ∈ V
83	75	fnmpt 6646	. . . . . . 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 6767	. . . . . 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 217	. . . . 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 10002	. . . . 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 8954	. . . 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 584	. . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)} ≼ (𝐴 ↑m 𝐵))
91		sbth 9044	. . 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 584	. 2 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})
93	7	brrelex2i 5694	. . . . 5 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
94	93	3ad2ant1 1133	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → 𝐴 ∈ V)
95		map0e 8827	. . . 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 8427	. . . . 5 ⊢ 1o ∈ V
98	97	enref 8932	. . . 4 ⊢ 1o ≈ 1o
99		df-sn 4592	. . . . 5 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
100		df1o2 8424	. . . . 5 ⊢ 1o = {∅}
101		en0 8964	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
102	101	anbi2i 623	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = ∅))
103		0ss 4361	. . . . . . . . 9 ⊢ ∅ ⊆ 𝐴
104		sseq1 3972	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
105	103, 104	mpbiri 257	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝑥 ⊆ 𝐴)
106	105	pm4.71ri 561	. . . . . . 7 ⊢ (𝑥 = ∅ ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 = ∅))
107	102, 106	bitr4i 277	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ 𝑥 = ∅)
108	107	abbii 2801	. . . . 5 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)} = {𝑥 ∣ 𝑥 = ∅}
109	99, 100, 108	3eqtr4ri 2770	. . . 4 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)} = 1o
110	98, 109	breqtrri 5137	. . 3 ⊢ 1o ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)}
111	96, 110	eqbrtrdi 5149	. 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → (𝐴 ↑m ∅) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)})
112	5, 92, 111	pm2.61ne 3026	1 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ≼ 𝐴 ∧ (𝐴 ↑m 𝐵) ∈ dom card) → (𝐴 ↑m 𝐵) ≈ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐵)})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2708   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  Vcvv 3446   ⊆ wss 3913  ∅c0 4287  {csn 4591   class class class wbr 5110   ↦ cmpt 5193   × cxp 5636  dom cdm 5638  ran crn 5639  Fun wfun 6495   Fn wfn 6496  ⟶wf 6497  –onto→wfo 6499  –1-1-onto→wf1o 6500  (class class class)co 7362  ωcom 7807  1_oc1o 8410   ↑_m cmap 8772   ≈ cen 8887   ≼ cdom 8888  cardccrd 9880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9586

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9455  df-card 9884  df-acn 9887

This theorem is referenced by:  infmap  10521