inmap - Mathbox for Glauco Siliprandi

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Theorem inmap 45661

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 4137	. . . . . . . . 9 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓 ∈ (𝐴 ↑_m 𝐶))
2		elmapi 8793	. . . . . . . . 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 4138	. . . . . . . . 9 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓 ∈ (𝐵 ↑_m 𝐶))
5		elmapi 8793	. . . . . . . . 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 516	. . . . . . 7 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → (𝑓:𝐶⟶𝐴 ∧ 𝑓:𝐶⟶𝐵))
8		fin 6714	. . . . . . 7 ⊢ (𝑓:𝐶⟶(𝐴 ∩ 𝐵) ↔ (𝑓:𝐶⟶𝐴 ∧ 𝑓:𝐶⟶𝐵))
9	7, 8	sylibr 235	. . . . . 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 4172	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
13	12	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐴)
14	11, 13	ssexd 5259	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
15		inmap.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
16	14, 15	elmapd 8784	. . . . . 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 258	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))) → 𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
19	18	ralrimiva 3132	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
20		dfss3 3911	. . 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 235	. 2 ⊢ (𝜑 → ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) ⊆ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
22		mapss 8834	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
23	11, 13, 22	syl2anc 590	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐴 ↑_m 𝐶))
24		inmap.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
25		inss2 4173	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
26	25	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
27		mapss 8834	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵) → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
28	24, 26, 27	syl2anc 590	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ (𝐵 ↑_m 𝐶))
29	23, 28	ssind 4176	. 2 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ↑_m 𝐶) ⊆ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)))
30	21, 29	eqssd 3939	1 ⊢ (𝜑 → ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) = ((𝐴 ∩ 𝐵) ↑_m 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∀wral 3054 Vcvv 3432 ∩ cin 3889 ⊆ wss 3890 ⟶wf 6488 (class class class)co 7363 ↑_m cmap 8770

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-map 8772

This theorem is referenced by: vonvolmbllem 47110 vonvolmbl 47111

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