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Theorem inmap 45102
Description: Intersection of two sets exponentiations. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
inmap.a (𝜑𝐴𝑉)
inmap.b (𝜑𝐵𝑊)
inmap.c (𝜑𝐶𝑍)
Assertion
Ref Expression
inmap (𝜑 → ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) = ((𝐴𝐵) ↑m 𝐶))

Proof of Theorem inmap
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 elinel1 4211 . . . . . . . . 9 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓 ∈ (𝐴m 𝐶))
2 elmapi 8882 . . . . . . . . 9 (𝑓 ∈ (𝐴m 𝐶) → 𝑓:𝐶𝐴)
31, 2syl 17 . . . . . . . 8 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓:𝐶𝐴)
4 elinel2 4212 . . . . . . . . 9 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓 ∈ (𝐵m 𝐶))
5 elmapi 8882 . . . . . . . . 9 (𝑓 ∈ (𝐵m 𝐶) → 𝑓:𝐶𝐵)
64, 5syl 17 . . . . . . . 8 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓:𝐶𝐵)
73, 6jca 511 . . . . . . 7 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → (𝑓:𝐶𝐴𝑓:𝐶𝐵))
8 fin 6783 . . . . . . 7 (𝑓:𝐶⟶(𝐴𝐵) ↔ (𝑓:𝐶𝐴𝑓:𝐶𝐵))
97, 8sylibr 234 . . . . . 6 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓:𝐶⟶(𝐴𝐵))
109adantl 481 . . . . 5 ((𝜑𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))) → 𝑓:𝐶⟶(𝐴𝐵))
11 inmap.a . . . . . . . 8 (𝜑𝐴𝑉)
12 inss1 4245 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
1312a1i 11 . . . . . . . 8 (𝜑 → (𝐴𝐵) ⊆ 𝐴)
1411, 13ssexd 5325 . . . . . . 7 (𝜑 → (𝐴𝐵) ∈ V)
15 inmap.c . . . . . . 7 (𝜑𝐶𝑍)
1614, 15elmapd 8873 . . . . . 6 (𝜑 → (𝑓 ∈ ((𝐴𝐵) ↑m 𝐶) ↔ 𝑓:𝐶⟶(𝐴𝐵)))
1716adantr 480 . . . . 5 ((𝜑𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))) → (𝑓 ∈ ((𝐴𝐵) ↑m 𝐶) ↔ 𝑓:𝐶⟶(𝐴𝐵)))
1810, 17mpbird 257 . . . 4 ((𝜑𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))) → 𝑓 ∈ ((𝐴𝐵) ↑m 𝐶))
1918ralrimiva 3142 . . 3 (𝜑 → ∀𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))𝑓 ∈ ((𝐴𝐵) ↑m 𝐶))
20 dfss3 3984 . . 3 (((𝐴m 𝐶) ∩ (𝐵m 𝐶)) ⊆ ((𝐴𝐵) ↑m 𝐶) ↔ ∀𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))𝑓 ∈ ((𝐴𝐵) ↑m 𝐶))
2119, 20sylibr 234 . 2 (𝜑 → ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) ⊆ ((𝐴𝐵) ↑m 𝐶))
22 mapss 8922 . . . 4 ((𝐴𝑉 ∧ (𝐴𝐵) ⊆ 𝐴) → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐴m 𝐶))
2311, 13, 22syl2anc 583 . . 3 (𝜑 → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐴m 𝐶))
24 inmap.b . . . 4 (𝜑𝐵𝑊)
25 inss2 4246 . . . . 5 (𝐴𝐵) ⊆ 𝐵
2625a1i 11 . . . 4 (𝜑 → (𝐴𝐵) ⊆ 𝐵)
27 mapss 8922 . . . 4 ((𝐵𝑊 ∧ (𝐴𝐵) ⊆ 𝐵) → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐵m 𝐶))
2824, 26, 27syl2anc 583 . . 3 (𝜑 → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐵m 𝐶))
2923, 28ssind 4249 . 2 (𝜑 → ((𝐴𝐵) ↑m 𝐶) ⊆ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)))
3021, 29eqssd 4013 1 (𝜑 → ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) = ((𝐴𝐵) ↑m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1535  wcel 2104  wral 3057  Vcvv 3477  cin 3962  wss 3963  wf 6554  (class class class)co 7425  m cmap 8859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5366  ax-pr 5430  ax-un 7747
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6510  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-1st 8007  df-2nd 8008  df-map 8861
This theorem is referenced by:  vonvolmbllem  46566  vonvolmbl  46567
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