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Theorem mapssbi 43912
Description: Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
mapssbi.a (𝜑𝐴𝑉)
mapssbi.b (𝜑𝐵𝑊)
mapssbi.c (𝜑𝐶𝑍)
mapssbi.n (𝜑𝐶 ≠ ∅)
Assertion
Ref Expression
mapssbi (𝜑 → (𝐴𝐵 ↔ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))

Proof of Theorem mapssbi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 mapssbi.b . . . . 5 (𝜑𝐵𝑊)
21adantr 482 . . . 4 ((𝜑𝐴𝐵) → 𝐵𝑊)
3 simpr 486 . . . 4 ((𝜑𝐴𝐵) → 𝐴𝐵)
4 mapss 8883 . . . 4 ((𝐵𝑊𝐴𝐵) → (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
52, 3, 4syl2anc 585 . . 3 ((𝜑𝐴𝐵) → (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
65ex 414 . 2 (𝜑 → (𝐴𝐵 → (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))
7 simplr 768 . . . 4 (((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ ¬ 𝐴𝐵) → (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
8 nssrex 43775 . . . . . . . 8 𝐴𝐵 ↔ ∃𝑥𝐴 ¬ 𝑥𝐵)
98biimpi 215 . . . . . . 7 𝐴𝐵 → ∃𝑥𝐴 ¬ 𝑥𝐵)
109adantl 483 . . . . . 6 ((𝜑 ∧ ¬ 𝐴𝐵) → ∃𝑥𝐴 ¬ 𝑥𝐵)
11 fconst6g 6781 . . . . . . . . . . . . 13 (𝑥𝐴 → (𝐶 × {𝑥}):𝐶𝐴)
1211adantl 483 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐶 × {𝑥}):𝐶𝐴)
13 mapssbi.a . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
14 mapssbi.c . . . . . . . . . . . . . 14 (𝜑𝐶𝑍)
15 elmapg 8833 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐶𝑍) → ((𝐶 × {𝑥}) ∈ (𝐴m 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1613, 14, 15syl2anc 585 . . . . . . . . . . . . 13 (𝜑 → ((𝐶 × {𝑥}) ∈ (𝐴m 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1716adantr 482 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝐶 × {𝑥}) ∈ (𝐴m 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1812, 17mpbird 257 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐶 × {𝑥}) ∈ (𝐴m 𝐶))
19183adant3 1133 . . . . . . . . . 10 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝐶 × {𝑥}) ∈ (𝐴m 𝐶))
2014adantr 482 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝐶𝑍)
211adantr 482 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝐵𝑊)
22 mapssbi.n . . . . . . . . . . . . . . 15 (𝜑𝐶 ≠ ∅)
2322adantr 482 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝐶 ≠ ∅)
24 simpr 486 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → (𝐶 × {𝑥}) ∈ (𝐵m 𝐶))
2520, 21, 23, 24snelmap 43771 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝑥𝐵)
2625adantlr 714 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑥𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝑥𝐵)
27 simplr 768 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑥𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → ¬ 𝑥𝐵)
2826, 27pm2.65da 816 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑥𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶))
29283adant2 1132 . . . . . . . . . 10 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶))
30 nelss 4048 . . . . . . . . . 10 (((𝐶 × {𝑥}) ∈ (𝐴m 𝐶) ∧ ¬ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
3119, 29, 30syl2anc 585 . . . . . . . . 9 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
32313exp 1120 . . . . . . . 8 (𝜑 → (𝑥𝐴 → (¬ 𝑥𝐵 → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))))
3332adantr 482 . . . . . . 7 ((𝜑 ∧ ¬ 𝐴𝐵) → (𝑥𝐴 → (¬ 𝑥𝐵 → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))))
3433rexlimdv 3154 . . . . . 6 ((𝜑 ∧ ¬ 𝐴𝐵) → (∃𝑥𝐴 ¬ 𝑥𝐵 → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))
3510, 34mpd 15 . . . . 5 ((𝜑 ∧ ¬ 𝐴𝐵) → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
3635adantlr 714 . . . 4 (((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ ¬ 𝐴𝐵) → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
377, 36condan 817 . . 3 ((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) → 𝐴𝐵)
3837ex 414 . 2 (𝜑 → ((𝐴m 𝐶) ⊆ (𝐵m 𝐶) → 𝐴𝐵))
396, 38impbid 211 1 (𝜑 → (𝐴𝐵 ↔ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  w3a 1088  wcel 2107  wne 2941  wrex 3071  wss 3949  c0 4323  {csn 4629   × cxp 5675  wf 6540  (class class class)co 7409  m cmap 8820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822
This theorem is referenced by: (None)
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