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Theorem mapssbi 42212
 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 484 . . . 4 ((𝜑𝐴𝐵) → 𝐵𝑊)
3 simpr 488 . . . 4 ((𝜑𝐴𝐵) → 𝐴𝐵)
4 mapss 8471 . . . 4 ((𝐵𝑊𝐴𝐵) → (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
52, 3, 4syl2anc 587 . . 3 ((𝜑𝐴𝐵) → (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
65ex 416 . 2 (𝜑 → (𝐴𝐵 → (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))
7 simplr 768 . . . 4 (((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ ¬ 𝐴𝐵) → (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
8 nssrex 42095 . . . . . . . 8 𝐴𝐵 ↔ ∃𝑥𝐴 ¬ 𝑥𝐵)
98biimpi 219 . . . . . . 7 𝐴𝐵 → ∃𝑥𝐴 ¬ 𝑥𝐵)
109adantl 485 . . . . . 6 ((𝜑 ∧ ¬ 𝐴𝐵) → ∃𝑥𝐴 ¬ 𝑥𝐵)
11 fconst6g 6553 . . . . . . . . . . . . 13 (𝑥𝐴 → (𝐶 × {𝑥}):𝐶𝐴)
1211adantl 485 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐶 × {𝑥}):𝐶𝐴)
13 mapssbi.a . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
14 mapssbi.c . . . . . . . . . . . . . 14 (𝜑𝐶𝑍)
15 elmapg 8429 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐶𝑍) → ((𝐶 × {𝑥}) ∈ (𝐴m 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1613, 14, 15syl2anc 587 . . . . . . . . . . . . 13 (𝜑 → ((𝐶 × {𝑥}) ∈ (𝐴m 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1716adantr 484 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝐶 × {𝑥}) ∈ (𝐴m 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1812, 17mpbird 260 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐶 × {𝑥}) ∈ (𝐴m 𝐶))
19183adant3 1129 . . . . . . . . . 10 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝐶 × {𝑥}) ∈ (𝐴m 𝐶))
2014adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝐶𝑍)
211adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝐵𝑊)
22 mapssbi.n . . . . . . . . . . . . . . 15 (𝜑𝐶 ≠ ∅)
2322adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝐶 ≠ ∅)
24 simpr 488 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → (𝐶 × {𝑥}) ∈ (𝐵m 𝐶))
2520, 21, 23, 24snelmap 42091 . . . . . . . . . . . . 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 1128 . . . . . . . . . 10 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶))
30 nelss 3955 . . . . . . . . . 10 (((𝐶 × {𝑥}) ∈ (𝐴m 𝐶) ∧ ¬ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
3119, 29, 30syl2anc 587 . . . . . . . . 9 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
32313exp 1116 . . . . . . . 8 (𝜑 → (𝑥𝐴 → (¬ 𝑥𝐵 → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))))
3332adantr 484 . . . . . . 7 ((𝜑 ∧ ¬ 𝐴𝐵) → (𝑥𝐴 → (¬ 𝑥𝐵 → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))))
3433rexlimdv 3207 . . . . . 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 416 . 2 (𝜑 → ((𝐴m 𝐶) ⊆ (𝐵m 𝐶) → 𝐴𝐵))
396, 38impbid 215 1 (𝜑 → (𝐴𝐵 ↔ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071   ⊆ wss 3858  ∅c0 4225  {csn 4522   × cxp 5522  ⟶wf 6331  (class class class)co 7150   ↑m cmap 8416 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-map 8418 This theorem is referenced by: (None)
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