inmap - Mathbox for Glauco Siliprandi

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Theorem inmap 45187

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 4154	. . . . . . . . 9 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓 ∈ (𝐴 ↑_m 𝐶))
2		elmapi 8783	. . . . . . . . 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 4155	. . . . . . . . 9 ⊢ (𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) → 𝑓 ∈ (𝐵 ↑_m 𝐶))
5		elmapi 8783	. . . . . . . . 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 6708	. . . . . . 7 ⊢ (𝑓:𝐶⟶(𝐴 ∩ 𝐵) ↔ (𝑓:𝐶⟶𝐴 ∧ 𝑓:𝐶⟶𝐵))
9	7, 8	sylibr 234	. . . . . 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 4190	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
13	12	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐴)
14	11, 13	ssexd 5266	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ V)
15		inmap.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑍)
16	14, 15	elmapd 8774	. . . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))) → 𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
19	18	ralrimiva 3121	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶))𝑓 ∈ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
20		dfss3 3926	. . 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 234	. 2 ⊢ (𝜑 → ((𝐴 ↑_m 𝐶) ∩ (𝐵 ↑_m 𝐶)) ⊆ ((𝐴 ∩ 𝐵) ↑_m 𝐶))
22		mapss 8823	. . . 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 4191	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
26	25	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
27		mapss 8823	. . . 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 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 Vcvv 3438 ∩ cin 3904 ⊆ wss 3905 ⟶wf 6482 (class class class)co 7353 ↑_m cmap 8760

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-fv 6494 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7931 df-2nd 7932 df-map 8762

This theorem is referenced by: vonvolmbllem 46642 vonvolmbl 46643

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