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Theorem mapssbi 43991
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 481 . . . 4 ((𝜑𝐴𝐵) → 𝐵𝑊)
3 simpr 485 . . . 4 ((𝜑𝐴𝐵) → 𝐴𝐵)
4 mapss 8885 . . . 4 ((𝐵𝑊𝐴𝐵) → (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
52, 3, 4syl2anc 584 . . 3 ((𝜑𝐴𝐵) → (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
65ex 413 . 2 (𝜑 → (𝐴𝐵 → (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))
7 simplr 767 . . . 4 (((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ ¬ 𝐴𝐵) → (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
8 nssrex 43857 . . . . . . . 8 𝐴𝐵 ↔ ∃𝑥𝐴 ¬ 𝑥𝐵)
98biimpi 215 . . . . . . 7 𝐴𝐵 → ∃𝑥𝐴 ¬ 𝑥𝐵)
109adantl 482 . . . . . 6 ((𝜑 ∧ ¬ 𝐴𝐵) → ∃𝑥𝐴 ¬ 𝑥𝐵)
11 fconst6g 6780 . . . . . . . . . . . . 13 (𝑥𝐴 → (𝐶 × {𝑥}):𝐶𝐴)
1211adantl 482 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐶 × {𝑥}):𝐶𝐴)
13 mapssbi.a . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
14 mapssbi.c . . . . . . . . . . . . . 14 (𝜑𝐶𝑍)
15 elmapg 8835 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐶𝑍) → ((𝐶 × {𝑥}) ∈ (𝐴m 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1613, 14, 15syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → ((𝐶 × {𝑥}) ∈ (𝐴m 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1716adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝐶 × {𝑥}) ∈ (𝐴m 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1812, 17mpbird 256 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐶 × {𝑥}) ∈ (𝐴m 𝐶))
19183adant3 1132 . . . . . . . . . 10 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝐶 × {𝑥}) ∈ (𝐴m 𝐶))
2014adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝐶𝑍)
211adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝐵𝑊)
22 mapssbi.n . . . . . . . . . . . . . . 15 (𝜑𝐶 ≠ ∅)
2322adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝐶 ≠ ∅)
24 simpr 485 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → (𝐶 × {𝑥}) ∈ (𝐵m 𝐶))
2520, 21, 23, 24snelmap 43853 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝑥𝐵)
2625adantlr 713 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑥𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝑥𝐵)
27 simplr 767 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑥𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → ¬ 𝑥𝐵)
2826, 27pm2.65da 815 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑥𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶))
29283adant2 1131 . . . . . . . . . 10 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶))
30 nelss 4047 . . . . . . . . . 10 (((𝐶 × {𝑥}) ∈ (𝐴m 𝐶) ∧ ¬ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
3119, 29, 30syl2anc 584 . . . . . . . . 9 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
32313exp 1119 . . . . . . . 8 (𝜑 → (𝑥𝐴 → (¬ 𝑥𝐵 → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))))
3332adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ 𝐴𝐵) → (𝑥𝐴 → (¬ 𝑥𝐵 → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))))
3433rexlimdv 3153 . . . . . 6 ((𝜑 ∧ ¬ 𝐴𝐵) → (∃𝑥𝐴 ¬ 𝑥𝐵 → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))
3510, 34mpd 15 . . . . 5 ((𝜑 ∧ ¬ 𝐴𝐵) → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
3635adantlr 713 . . . 4 (((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ ¬ 𝐴𝐵) → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
377, 36condan 816 . . 3 ((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) → 𝐴𝐵)
3837ex 413 . 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 396  w3a 1087  wcel 2106  wne 2940  wrex 3070  wss 3948  c0 4322  {csn 4628   × cxp 5674  wf 6539  (class class class)co 7411  m cmap 8822
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824
This theorem is referenced by: (None)
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