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Theorem inmap 41831
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 4125 . . . . . . . . 9 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓 ∈ (𝐴m 𝐶))
2 elmapi 8415 . . . . . . . . 9 (𝑓 ∈ (𝐴m 𝐶) → 𝑓:𝐶𝐴)
31, 2syl 17 . . . . . . . 8 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓:𝐶𝐴)
4 elinel2 4126 . . . . . . . . 9 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓 ∈ (𝐵m 𝐶))
5 elmapi 8415 . . . . . . . . 9 (𝑓 ∈ (𝐵m 𝐶) → 𝑓:𝐶𝐵)
64, 5syl 17 . . . . . . . 8 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓:𝐶𝐵)
73, 6jca 515 . . . . . . 7 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → (𝑓:𝐶𝐴𝑓:𝐶𝐵))
8 fin 6537 . . . . . . 7 (𝑓:𝐶⟶(𝐴𝐵) ↔ (𝑓:𝐶𝐴𝑓:𝐶𝐵))
97, 8sylibr 237 . . . . . 6 (𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) → 𝑓:𝐶⟶(𝐴𝐵))
109adantl 485 . . . . 5 ((𝜑𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))) → 𝑓:𝐶⟶(𝐴𝐵))
11 inmap.a . . . . . . . 8 (𝜑𝐴𝑉)
12 inss1 4158 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
1312a1i 11 . . . . . . . 8 (𝜑 → (𝐴𝐵) ⊆ 𝐴)
1411, 13ssexd 5195 . . . . . . 7 (𝜑 → (𝐴𝐵) ∈ V)
15 inmap.c . . . . . . 7 (𝜑𝐶𝑍)
1614, 15elmapd 8407 . . . . . 6 (𝜑 → (𝑓 ∈ ((𝐴𝐵) ↑m 𝐶) ↔ 𝑓:𝐶⟶(𝐴𝐵)))
1716adantr 484 . . . . 5 ((𝜑𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))) → (𝑓 ∈ ((𝐴𝐵) ↑m 𝐶) ↔ 𝑓:𝐶⟶(𝐴𝐵)))
1810, 17mpbird 260 . . . 4 ((𝜑𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))) → 𝑓 ∈ ((𝐴𝐵) ↑m 𝐶))
1918ralrimiva 3152 . . 3 (𝜑 → ∀𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))𝑓 ∈ ((𝐴𝐵) ↑m 𝐶))
20 dfss3 3906 . . 3 (((𝐴m 𝐶) ∩ (𝐵m 𝐶)) ⊆ ((𝐴𝐵) ↑m 𝐶) ↔ ∀𝑓 ∈ ((𝐴m 𝐶) ∩ (𝐵m 𝐶))𝑓 ∈ ((𝐴𝐵) ↑m 𝐶))
2119, 20sylibr 237 . 2 (𝜑 → ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) ⊆ ((𝐴𝐵) ↑m 𝐶))
22 mapss 8440 . . . 4 ((𝐴𝑉 ∧ (𝐴𝐵) ⊆ 𝐴) → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐴m 𝐶))
2311, 13, 22syl2anc 587 . . 3 (𝜑 → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐴m 𝐶))
24 inmap.b . . . 4 (𝜑𝐵𝑊)
25 inss2 4159 . . . . 5 (𝐴𝐵) ⊆ 𝐵
2625a1i 11 . . . 4 (𝜑 → (𝐴𝐵) ⊆ 𝐵)
27 mapss 8440 . . . 4 ((𝐵𝑊 ∧ (𝐴𝐵) ⊆ 𝐵) → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐵m 𝐶))
2824, 26, 27syl2anc 587 . . 3 (𝜑 → ((𝐴𝐵) ↑m 𝐶) ⊆ (𝐵m 𝐶))
2923, 28ssind 4162 . 2 (𝜑 → ((𝐴𝐵) ↑m 𝐶) ⊆ ((𝐴m 𝐶) ∩ (𝐵m 𝐶)))
3021, 29eqssd 3935 1 (𝜑 → ((𝐴m 𝐶) ∩ (𝐵m 𝐶)) = ((𝐴𝐵) ↑m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  Vcvv 3444  cin 3883  wss 3884  wf 6324  (class class class)co 7139  m cmap 8393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395
This theorem is referenced by:  vonvolmbllem  43292  vonvolmbl  43293
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