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Theorem mapss2 45148
Description: Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
mapss2.a (𝜑𝐴𝑉)
mapss2.b (𝜑𝐵𝑊)
mapss2.c (𝜑𝐶𝑍)
mapss2.n (𝜑𝐶 ≠ ∅)
Assertion
Ref Expression
mapss2 (𝜑 → (𝐴𝐵 ↔ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))

Proof of Theorem mapss2
Dummy variables 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mapss2.b . . . . 5 (𝜑𝐵𝑊)
21adantr 480 . . . 4 ((𝜑𝐴𝐵) → 𝐵𝑊)
3 simpr 484 . . . 4 ((𝜑𝐴𝐵) → 𝐴𝐵)
4 mapss 8928 . . . 4 ((𝐵𝑊𝐴𝐵) → (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
52, 3, 4syl2anc 584 . . 3 ((𝜑𝐴𝐵) → (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
65ex 412 . 2 (𝜑 → (𝐴𝐵 → (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))
7 mapss2.n . . . . . 6 (𝜑𝐶 ≠ ∅)
8 n0 4359 . . . . . 6 (𝐶 ≠ ∅ ↔ ∃𝑥 𝑥𝐶)
97, 8sylib 218 . . . . 5 (𝜑 → ∃𝑥 𝑥𝐶)
109adantr 480 . . . 4 ((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) → ∃𝑥 𝑥𝐶)
11 eqidd 2736 . . . . . . . . . . . 12 ((𝜑𝑥𝐶) → (𝑤𝐶𝑦) = (𝑤𝐶𝑦))
12 eqidd 2736 . . . . . . . . . . . 12 (((𝜑𝑥𝐶) ∧ 𝑤 = 𝑥) → 𝑦 = 𝑦)
13 simpr 484 . . . . . . . . . . . 12 ((𝜑𝑥𝐶) → 𝑥𝐶)
14 vex 3482 . . . . . . . . . . . . 13 𝑦 ∈ V
1514a1i 11 . . . . . . . . . . . 12 ((𝜑𝑥𝐶) → 𝑦 ∈ V)
1611, 12, 13, 15fvmptd 7023 . . . . . . . . . . 11 ((𝜑𝑥𝐶) → ((𝑤𝐶𝑦)‘𝑥) = 𝑦)
1716eqcomd 2741 . . . . . . . . . 10 ((𝜑𝑥𝐶) → 𝑦 = ((𝑤𝐶𝑦)‘𝑥))
1817ad4ant13 751 . . . . . . . . 9 ((((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ 𝑥𝐶) ∧ 𝑦𝐴) → 𝑦 = ((𝑤𝐶𝑦)‘𝑥))
19 simplr 769 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ 𝑦𝐴) → (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
20 simplr 769 . . . . . . . . . . . . . . . 16 (((𝜑𝑦𝐴) ∧ 𝑤𝐶) → 𝑦𝐴)
2120fmpttd 7135 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝐴) → (𝑤𝐶𝑦):𝐶𝐴)
22 mapss2.a . . . . . . . . . . . . . . . . 17 (𝜑𝐴𝑉)
2322adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐴) → 𝐴𝑉)
24 mapss2.c . . . . . . . . . . . . . . . . 17 (𝜑𝐶𝑍)
2524adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝐴) → 𝐶𝑍)
2623, 25elmapd 8879 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝐴) → ((𝑤𝐶𝑦) ∈ (𝐴m 𝐶) ↔ (𝑤𝐶𝑦):𝐶𝐴))
2721, 26mpbird 257 . . . . . . . . . . . . . 14 ((𝜑𝑦𝐴) → (𝑤𝐶𝑦) ∈ (𝐴m 𝐶))
2827adantlr 715 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ 𝑦𝐴) → (𝑤𝐶𝑦) ∈ (𝐴m 𝐶))
2919, 28sseldd 3996 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ 𝑦𝐴) → (𝑤𝐶𝑦) ∈ (𝐵m 𝐶))
30 elmapi 8888 . . . . . . . . . . . 12 ((𝑤𝐶𝑦) ∈ (𝐵m 𝐶) → (𝑤𝐶𝑦):𝐶𝐵)
3129, 30syl 17 . . . . . . . . . . 11 (((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ 𝑦𝐴) → (𝑤𝐶𝑦):𝐶𝐵)
3231adantlr 715 . . . . . . . . . 10 ((((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ 𝑥𝐶) ∧ 𝑦𝐴) → (𝑤𝐶𝑦):𝐶𝐵)
33 simplr 769 . . . . . . . . . 10 ((((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ 𝑥𝐶) ∧ 𝑦𝐴) → 𝑥𝐶)
3432, 33ffvelcdmd 7105 . . . . . . . . 9 ((((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ 𝑥𝐶) ∧ 𝑦𝐴) → ((𝑤𝐶𝑦)‘𝑥) ∈ 𝐵)
3518, 34eqeltrd 2839 . . . . . . . 8 ((((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ 𝑥𝐶) ∧ 𝑦𝐴) → 𝑦𝐵)
3635ralrimiva 3144 . . . . . . 7 (((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ 𝑥𝐶) → ∀𝑦𝐴 𝑦𝐵)
37 dfss3 3984 . . . . . . 7 (𝐴𝐵 ↔ ∀𝑦𝐴 𝑦𝐵)
3836, 37sylibr 234 . . . . . 6 (((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ 𝑥𝐶) → 𝐴𝐵)
3938ex 412 . . . . 5 ((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) → (𝑥𝐶𝐴𝐵))
4039exlimdv 1931 . . . 4 ((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) → (∃𝑥 𝑥𝐶𝐴𝐵))
4110, 40mpd 15 . . 3 ((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) → 𝐴𝐵)
4241ex 412 . 2 (𝜑 → ((𝐴m 𝐶) ⊆ (𝐵m 𝐶) → 𝐴𝐵))
436, 42impbid 212 1 (𝜑 → (𝐴𝐵 ↔ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  Vcvv 3478  wss 3963  c0 4339  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867
This theorem is referenced by:  ovnovollem1  46612  ovnovollem2  46613
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