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Theorem mapssbi 40157
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 473 . . . 4 ((𝜑𝐴𝐵) → 𝐵𝑊)
3 simpr 478 . . . 4 ((𝜑𝐴𝐵) → 𝐴𝐵)
4 mapss 8140 . . . 4 ((𝐵𝑊𝐴𝐵) → (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
52, 3, 4syl2anc 580 . . 3 ((𝜑𝐴𝐵) → (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
65ex 402 . 2 (𝜑 → (𝐴𝐵 → (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)))
7 simplr 786 . . . 4 (((𝜑 ∧ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)) ∧ ¬ 𝐴𝐵) → (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
8 nssrex 40019 . . . . . . . 8 𝐴𝐵 ↔ ∃𝑥𝐴 ¬ 𝑥𝐵)
98biimpi 208 . . . . . . 7 𝐴𝐵 → ∃𝑥𝐴 ¬ 𝑥𝐵)
109adantl 474 . . . . . 6 ((𝜑 ∧ ¬ 𝐴𝐵) → ∃𝑥𝐴 ¬ 𝑥𝐵)
11 fconst6g 6309 . . . . . . . . . . . . 13 (𝑥𝐴 → (𝐶 × {𝑥}):𝐶𝐴)
1211adantl 474 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐶 × {𝑥}):𝐶𝐴)
13 mapssbi.a . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
14 mapssbi.c . . . . . . . . . . . . . 14 (𝜑𝐶𝑍)
15 elmapg 8108 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐶𝑍) → ((𝐶 × {𝑥}) ∈ (𝐴𝑚 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1613, 14, 15syl2anc 580 . . . . . . . . . . . . 13 (𝜑 → ((𝐶 × {𝑥}) ∈ (𝐴𝑚 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1716adantr 473 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝐶 × {𝑥}) ∈ (𝐴𝑚 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1812, 17mpbird 249 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐶 × {𝑥}) ∈ (𝐴𝑚 𝐶))
19183adant3 1163 . . . . . . . . . 10 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → (𝐶 × {𝑥}) ∈ (𝐴𝑚 𝐶))
2014adantr 473 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶)) → 𝐶𝑍)
211adantr 473 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶)) → 𝐵𝑊)
22 mapssbi.n . . . . . . . . . . . . . . 15 (𝜑𝐶 ≠ ∅)
2322adantr 473 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶)) → 𝐶 ≠ ∅)
24 simpr 478 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶)) → (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶))
2520, 21, 23, 24snelmap 40013 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶)) → 𝑥𝐵)
2625adantlr 707 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑥𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶)) → 𝑥𝐵)
27 simplr 786 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑥𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶)) → ¬ 𝑥𝐵)
2826, 27pm2.65da 852 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑥𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶))
29283adant2 1162 . . . . . . . . . 10 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶))
30 nelss 3860 . . . . . . . . . 10 (((𝐶 × {𝑥}) ∈ (𝐴𝑚 𝐶) ∧ ¬ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶)) → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
3119, 29, 30syl2anc 580 . . . . . . . . 9 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
32313exp 1149 . . . . . . . 8 (𝜑 → (𝑥𝐴 → (¬ 𝑥𝐵 → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))))
3332adantr 473 . . . . . . 7 ((𝜑 ∧ ¬ 𝐴𝐵) → (𝑥𝐴 → (¬ 𝑥𝐵 → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))))
3433rexlimdv 3211 . . . . . 6 ((𝜑 ∧ ¬ 𝐴𝐵) → (∃𝑥𝐴 ¬ 𝑥𝐵 → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)))
3510, 34mpd 15 . . . . 5 ((𝜑 ∧ ¬ 𝐴𝐵) → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
3635adantlr 707 . . . 4 (((𝜑 ∧ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)) ∧ ¬ 𝐴𝐵) → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
377, 36condan 853 . . 3 ((𝜑 ∧ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)) → 𝐴𝐵)
3837ex 402 . 2 (𝜑 → ((𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶) → 𝐴𝐵))
396, 38impbid 204 1 (𝜑 → (𝐴𝐵 ↔ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  w3a 1108  wcel 2157  wne 2971  wrex 3090  wss 3769  c0 4115  {csn 4368   × cxp 5310  wf 6097  (class class class)co 6878  𝑚 cmap 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-map 8097
This theorem is referenced by: (None)
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