inmap - Mathbox for Glauco Siliprandi

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Theorem inmap 45905

Description: Intersection of two sets exponentiations. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
inmap.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
inmap.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
inmap.c	⊢ (𝜑 → 𝐶 ∈ 𝑍)

**Assertion**
Ref	Expression
inmap	⊢ (𝜑 → ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) = ((𝐴 ∩ 𝐵) ↑_m 𝐶))

Proof of Theorem inmap Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elinel1 4155	. . . . . . . . 9 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓 ∈ (𝐴 ↑_m 𝐶))
2		elmapi 8847	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝐴 ↑_m 𝐶) → 𝑓:𝐶⟶𝐴)
3	1, 2	syl 18	. . . . . . . 8 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓:𝐶⟶𝐴)
4		elinel2 4156	. . . . . . . . 9 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓 ∈ (𝐵 ↑_m 𝐶))
5		elmapi 8847	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝐵 ↑_m 𝐶) → 𝑓:𝐶⟶𝐵)
6	4, 5	syl 18	. . . . . . . 8 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓:𝐶⟶𝐵)
7	3, 6	jca 520	. . . . . . 7 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → (𝑓:𝐶⟶𝐴 ∧ 𝑓:𝐶⟶𝐵))
8		fin 6760	. . . . . . 7 ⊢ (𝑓:𝐶⟶(𝐴 ∩ 𝐵) ↔ (𝑓:𝐶⟶𝐴 ∧ 𝑓:𝐶⟶𝐵))
9	7, 8	sylibr 237	. . . . . 6 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓:𝐶⟶(𝐴 ∩ 𝐵))
10	9	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))) → 𝑓:𝐶⟶(𝐴 ∩ 𝐵))
11		inmap.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
12		inss1 4190	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
13	12	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐴)
14	11, 13	ssexd 5296	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
15		inmap.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
16	14, 15	elmapd 8838	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶) ↔ 𝑓:𝐶⟶(𝐴 ∩ 𝐵)))
17	16	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))) → (𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶) ↔ 𝑓:𝐶⟶(𝐴 ∩ 𝐵)))
18	10, 17	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))) → 𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
19	18	ralrimiva 3157	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
20		dfss3 3927	. . 3 ⊢ (((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) ⊆ ((𝐴 ∩ 𝐵) ↑_m 𝐶) ↔ ∀𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
21	19, 20	sylibr 237	. 2 ⊢ (𝜑 → ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) ⊆ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
22		mapss 8888	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
23	11, 13, 22	syl2anc 595	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
24		inmap.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
25		inss2 4191	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
26	25	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
27		mapss 8888	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
28	24, 26, 27	syl2anc 595	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
29	23, 28	ssind 4194	. 2 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)))
30	21, 29	eqssd 3955	1 ⊢ (𝜑 → ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) = ((𝐴 ∩ 𝐵) ↑_m 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 Vcvv 3455 ∩ cin 3905 ⊆ wss 3906 ⟶wf 6534 (class class class)co 7412 ↑_m cmap 8825

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-fv 6546 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7987 df-2nd 7988 df-map 8827

This theorem is referenced by: vonvolmbllem 47354 vonvolmbl 47355

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