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Theorem mapssbi 41341
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 8442 . . . 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 765 . . . 4 (((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ ¬ 𝐴𝐵) → (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
8 nssrex 41216 . . . . . . . 8 𝐴𝐵 ↔ ∃𝑥𝐴 ¬ 𝑥𝐵)
98biimpi 217 . . . . . . 7 𝐴𝐵 → ∃𝑥𝐴 ¬ 𝑥𝐵)
109adantl 482 . . . . . 6 ((𝜑 ∧ ¬ 𝐴𝐵) → ∃𝑥𝐴 ¬ 𝑥𝐵)
11 fconst6g 6565 . . . . . . . . . . . . 13 (𝑥𝐴 → (𝐶 × {𝑥}):𝐶𝐴)
1211adantl 482 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐶 × {𝑥}):𝐶𝐴)
13 mapssbi.a . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
14 mapssbi.c . . . . . . . . . . . . . 14 (𝜑𝐶𝑍)
15 elmapg 8409 . . . . . . . . . . . . . 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 258 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐶 × {𝑥}) ∈ (𝐴m 𝐶))
19183adant3 1126 . . . . . . . . . 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 41211 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝑥𝐵)
2625adantlr 711 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑥𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → 𝑥𝐵)
27 simplr 765 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑥𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶)) → ¬ 𝑥𝐵)
2826, 27pm2.65da 813 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑥𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶))
29283adant2 1125 . . . . . . . . . 10 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵m 𝐶))
30 nelss 4034 . . . . . . . . . 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 1113 . . . . . . . 8 (𝜑 → (𝑥𝐴 → (¬ 𝑥𝐵 → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))))
3332adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ 𝐴𝐵) → (𝑥𝐴 → (¬ 𝑥𝐵 → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))))
3433rexlimdv 3288 . . . . . 6 ((𝜑 ∧ ¬ 𝐴𝐵) → (∃𝑥𝐴 ¬ 𝑥𝐵 → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))
3510, 34mpd 15 . . . . 5 ((𝜑 ∧ ¬ 𝐴𝐵) → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
3635adantlr 711 . . . 4 (((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) ∧ ¬ 𝐴𝐵) → ¬ (𝐴m 𝐶) ⊆ (𝐵m 𝐶))
377, 36condan 814 . . 3 ((𝜑 ∧ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)) → 𝐴𝐵)
3837ex 413 . 2 (𝜑 → ((𝐴m 𝐶) ⊆ (𝐵m 𝐶) → 𝐴𝐵))
396, 38impbid 213 1 (𝜑 → (𝐴𝐵 ↔ (𝐴m 𝐶) ⊆ (𝐵m 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081  wcel 2107  wne 3021  wrex 3144  wss 3940  c0 4295  {csn 4564   × cxp 5552  wf 6348  (class class class)co 7148  m cmap 8396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-map 8398
This theorem is referenced by: (None)
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