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Theorem inmap 41348
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 4169 . . . . . . . . 9 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓 ∈ (𝐴m 𝐶))
2 elmapi 8417 . . . . . . . . 9 (𝑓 ∈ (𝐴m 𝐶) → 𝑓:𝐶𝐴)
31, 2syl 17 . . . . . . . 8 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓:𝐶𝐴)
4 elinel2 4170 . . . . . . . . 9 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓 ∈ (𝐵m 𝐶))
5 elmapi 8417 . . . . . . . . 9 (𝑓 ∈ (𝐵m 𝐶) → 𝑓:𝐶𝐵)
64, 5syl 17 . . . . . . . 8 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓:𝐶𝐵)
73, 6jca 512 . . . . . . 7 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → (𝑓:𝐶𝐴𝑓:𝐶𝐵))
8 fin 6552 . . . . . . 7 (𝑓:𝐶⟶(𝐴𝐵) ↔ (𝑓:𝐶𝐴𝑓:𝐶𝐵))
97, 8sylibr 235 . . . . . 6 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓:𝐶⟶(𝐴𝐵))
109adantl 482 . . . . 5 ((𝜑𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))) → 𝑓:𝐶⟶(𝐴𝐵))
11 inmap.a . . . . . . . 8 (𝜑𝐴𝑉)
12 inss1 4202 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
1312a1i 11 . . . . . . . 8 (𝜑 → (𝐴𝐵) ⊆ 𝐴)
1411, 13ssexd 5219 . . . . . . 7 (𝜑 → (𝐴𝐵) ∈ V)
15 inmap.c . . . . . . 7 (𝜑𝐶𝑍)
1614, 15elmapd 8409 . . . . . 6 (𝜑 → (𝑓 ∈ ((𝐴𝐵) ↑m 𝐶) ↔ 𝑓:𝐶⟶(𝐴𝐵)))
1716adantr 481 . . . . 5 ((𝜑𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))) → (𝑓 ∈ ((𝐴𝐵) ↑m 𝐶) ↔ 𝑓:𝐶⟶(𝐴𝐵)))
1810, 17mpbird 258 . . . 4 ((𝜑𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))) → 𝑓 ∈ ((𝐴𝐵) ↑m 𝐶))
1918ralrimiva 3179 . . 3 (𝜑 → ∀𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))𝑓 ∈ ((𝐴𝐵) ↑m 𝐶))
20 dfss3 3953 . . 3 (((𝐴m 𝐶) ∩ (𝐵m 𝐶)) ⊆ ((𝐴𝐵) ↑m 𝐶) ↔ ∀𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))𝑓 ∈ ((𝐴𝐵) ↑m 𝐶))
2119, 20sylibr 235 . 2 (𝜑 → ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) ⊆ ((𝐴𝐵) ↑m 𝐶))
22 mapss 8441 . . . 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 4203 . . . . 5 (𝐴𝐵) ⊆ 𝐵
2625a1i 11 . . . 4 (𝜑 → (𝐴𝐵) ⊆ 𝐵)
27 mapss 8441 . . . 4 ((𝐵𝑊 ∧ (𝐴𝐵) ⊆ 𝐵) → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐵m 𝐶))
2824, 26, 27syl2anc 584 . . 3 (𝜑 → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐵m 𝐶))
2923, 28ssind 4206 . 2 (𝜑 → ((𝐴𝐵) ↑m 𝐶) ⊆ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)))
3021, 29eqssd 3981 1 (𝜑 → ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) = ((𝐴𝐵) ↑m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cin 3932  wss 3933  wf 6344  (class class class)co 7145  m cmap 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397
This theorem is referenced by:  vonvolmbllem  42819  vonvolmbl  42820
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