inmap - Mathbox for Glauco Siliprandi

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Theorem inmap 42638

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 4125	. . . . . . . . 9 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓 ∈ (𝐴 ↑_m 𝐶))
2		elmapi 8595	. . . . . . . . 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 4126	. . . . . . . . 9 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓 ∈ (𝐵 ↑_m 𝐶))
5		elmapi 8595	. . . . . . . . 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 511	. . . . . . 7 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → (𝑓:𝐶⟶𝐴 ∧ 𝑓:𝐶⟶𝐵))
8		fin 6638	. . . . . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))) → 𝑓:𝐶⟶(𝐴 ∩ 𝐵))
11		inmap.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
12		inss1 4159	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
13	12	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐴)
14	11, 13	ssexd 5243	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
15		inmap.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
16	14, 15	elmapd 8587	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶) ↔ 𝑓:𝐶⟶(𝐴 ∩ 𝐵)))
17	16	adantr 480	. . . . 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 3107	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
20		dfss3 3905	. . 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 8635	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
23	11, 13, 22	syl2anc 583	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
24		inmap.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
25		inss2 4160	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
26	25	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
27		mapss 8635	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
28	24, 26, 27	syl2anc 583	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
29	23, 28	ssind 4163	. 2 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)))
30	21, 29	eqssd 3934	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 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 Vcvv 3422 ∩ cin 3882 ⊆ wss 3883 ⟶wf 6414 (class class class)co 7255 ↑_m cmap 8573

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-map 8575

This theorem is referenced by: vonvolmbllem 44088 vonvolmbl 44089

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