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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 (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 4200 . . . . . . . . 9 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓 ∈ (𝐴m 𝐶))
2 elmapi 8885 . . . . . . . . 9 (𝑓 ∈ (𝐴m 𝐶) → 𝑓:𝐶𝐴)
31, 2syl 17 . . . . . . . 8 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓:𝐶𝐴)
4 elinel2 4201 . . . . . . . . 9 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓 ∈ (𝐵m 𝐶))
5 elmapi 8885 . . . . . . . . 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 6786 . . . . . . 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 4236 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
1312a1i 11 . . . . . . . 8 (𝜑 → (𝐴𝐵) ⊆ 𝐴)
1411, 13ssexd 5322 . . . . . . 7 (𝜑 → (𝐴𝐵) ∈ V)
15 inmap.c . . . . . . 7 (𝜑𝐶𝑍)
1614, 15elmapd 8876 . . . . . 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 3145 . . 3 (𝜑 → ∀𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))𝑓 ∈ ((𝐴𝐵) ↑m 𝐶))
20 dfss3 3971 . . 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 8925 . . . 4 ((𝐴𝑉 ∧ (𝐴𝐵) ⊆ 𝐴) → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐴m 𝐶))
2311, 13, 22syl2anc 584 . . 3 (𝜑 → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐴m 𝐶))
24 inmap.b . . . 4 (𝜑𝐵𝑊)
25 inss2 4237 . . . . 5 (𝐴𝐵) ⊆ 𝐵
2625a1i 11 . . . 4 (𝜑 → (𝐴𝐵) ⊆ 𝐵)
27 mapss 8925 . . . 4 ((𝐵𝑊 ∧ (𝐴𝐵) ⊆ 𝐵) → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐵m 𝐶))
2824, 26, 27syl2anc 584 . . 3 (𝜑 → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐵m 𝐶))
2923, 28ssind 4240 . 2 (𝜑 → ((𝐴𝐵) ↑m 𝐶) ⊆ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)))
3021, 29eqssd 4000 1 (𝜑 → ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) = ((𝐴𝐵) ↑m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3060  Vcvv 3479  cin 3949  wss 3950  wf 6555  (class class class)co 7429  m cmap 8862
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  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 6512  df-fun 6561  df-fn 6562  df-f 6563  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-map 8864
This theorem is referenced by:  vonvolmbllem  46648  vonvolmbl  46649
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