inmap - Mathbox for Glauco Siliprandi

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Theorem inmap 43128

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 4143	. . . . . . . . 9 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓 ∈ (𝐴 ↑_m 𝐶))
2		elmapi 8709	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝐴 ↑_m 𝐶) → 𝑓:𝐶⟶𝐴)
3	1, 2	syl 17	. . . . . . . 8 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓:𝐶⟶𝐴)
4		elinel2 4144	. . . . . . . . 9 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓 ∈ (𝐵 ↑_m 𝐶))
5		elmapi 8709	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝐵 ↑_m 𝐶) → 𝑓:𝐶⟶𝐵)
6	4, 5	syl 17	. . . . . . . 8 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓:𝐶⟶𝐵)
7	3, 6	jca 512	. . . . . . 7 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → (𝑓:𝐶⟶𝐴 ∧ 𝑓:𝐶⟶𝐵))
8		fin 6706	. . . . . . 7 ⊢ (𝑓:𝐶⟶(𝐴 ∩ 𝐵) ↔ (𝑓:𝐶⟶𝐴 ∧ 𝑓:𝐶⟶𝐵))
9	7, 8	sylibr 233	. . . . . 6 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓:𝐶⟶(𝐴 ∩ 𝐵))
10	9	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))) → 𝑓:𝐶⟶(𝐴 ∩ 𝐵))
11		inmap.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
12		inss1 4176	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
13	12	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐴)
14	11, 13	ssexd 5269	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
15		inmap.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
16	14, 15	elmapd 8701	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶) ↔ 𝑓:𝐶⟶(𝐴 ∩ 𝐵)))
17	16	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))) → (𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶) ↔ 𝑓:𝐶⟶(𝐴 ∩ 𝐵)))
18	10, 17	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))) → 𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
19	18	ralrimiva 3139	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
20		dfss3 3920	. . 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 233	. 2 ⊢ (𝜑 → ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) ⊆ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
22		mapss 8749	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
23	11, 13, 22	syl2anc 584	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
24		inmap.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
25		inss2 4177	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
26	25	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
27		mapss 8749	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
28	24, 26, 27	syl2anc 584	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
29	23, 28	ssind 4180	. 2 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)))
30	21, 29	eqssd 3949	1 ⊢ (𝜑 → ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) = ((𝐴 ∩ 𝐵) ↑_m 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ∀wral 3061 Vcvv 3441 ∩ cin 3897 ⊆ wss 3898 ⟶wf 6476 (class class class)co 7338 ↑_m cmap 8687

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-iun 4944 df-br 5094 df-opab 5156 df-mpt 5177 df-id 5519 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-fv 6488 df-ov 7341 df-oprab 7342 df-mpo 7343 df-1st 7900 df-2nd 7901 df-map 8689

This theorem is referenced by: vonvolmbllem 44587 vonvolmbl 44588

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