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Theorem mapssbi 38910
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 477 . . . 4 ((𝜑𝐴𝐵) → 𝐴𝐵)
4 mapss 7852 . . . 4 ((𝐵𝑊𝐴𝐵) → (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
52, 3, 4syl2anc 692 . . 3 ((𝜑𝐴𝐵) → (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
65ex 450 . 2 (𝜑 → (𝐴𝐵 → (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)))
7 simplr 791 . . . 4 (((𝜑 ∧ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)) ∧ ¬ 𝐴𝐵) → (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
8 nssrex 38778 . . . . . . . 8 𝐴𝐵 ↔ ∃𝑥𝐴 ¬ 𝑥𝐵)
98biimpi 206 . . . . . . 7 𝐴𝐵 → ∃𝑥𝐴 ¬ 𝑥𝐵)
109adantl 482 . . . . . 6 ((𝜑 ∧ ¬ 𝐴𝐵) → ∃𝑥𝐴 ¬ 𝑥𝐵)
11 fconst6g 6056 . . . . . . . . . . . . 13 (𝑥𝐴 → (𝐶 × {𝑥}):𝐶𝐴)
1211adantl 482 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → (𝐶 × {𝑥}):𝐶𝐴)
13 mapssbi.a . . . . . . . . . . . . . 14 (𝜑𝐴𝑉)
14 mapssbi.c . . . . . . . . . . . . . 14 (𝜑𝐶𝑍)
15 elmapg 7822 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐶𝑍) → ((𝐶 × {𝑥}) ∈ (𝐴𝑚 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1613, 14, 15syl2anc 692 . . . . . . . . . . . . 13 (𝜑 → ((𝐶 × {𝑥}) ∈ (𝐴𝑚 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1716adantr 481 . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → ((𝐶 × {𝑥}) ∈ (𝐴𝑚 𝐶) ↔ (𝐶 × {𝑥}):𝐶𝐴))
1812, 17mpbird 247 . . . . . . . . . . 11 ((𝜑𝑥𝐴) → (𝐶 × {𝑥}) ∈ (𝐴𝑚 𝐶))
19183adant3 1079 . . . . . . . . . 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 477 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶)) → (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶))
2520, 21, 23, 24snelmap 38772 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶)) → 𝑥𝐵)
2625adantlr 750 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑥𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶)) → 𝑥𝐵)
27 simplr 791 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑥𝐵) ∧ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶)) → ¬ 𝑥𝐵)
2826, 27pm2.65da 599 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝑥𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶))
29283adant2 1078 . . . . . . . . . 10 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶))
30 nelss 3648 . . . . . . . . . 10 (((𝐶 × {𝑥}) ∈ (𝐴𝑚 𝐶) ∧ ¬ (𝐶 × {𝑥}) ∈ (𝐵𝑚 𝐶)) → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
3119, 29, 30syl2anc 692 . . . . . . . . 9 ((𝜑𝑥𝐴 ∧ ¬ 𝑥𝐵) → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
32313exp 1261 . . . . . . . 8 (𝜑 → (𝑥𝐴 → (¬ 𝑥𝐵 → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))))
3332adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ 𝐴𝐵) → (𝑥𝐴 → (¬ 𝑥𝐵 → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))))
3433rexlimdv 3024 . . . . . 6 ((𝜑 ∧ ¬ 𝐴𝐵) → (∃𝑥𝐴 ¬ 𝑥𝐵 → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)))
3510, 34mpd 15 . . . . 5 ((𝜑 ∧ ¬ 𝐴𝐵) → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
3635adantlr 750 . . . 4 (((𝜑 ∧ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)) ∧ ¬ 𝐴𝐵) → ¬ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶))
377, 36condan 834 . . 3 ((𝜑 ∧ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)) → 𝐴𝐵)
3837ex 450 . 2 (𝜑 → ((𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶) → 𝐴𝐵))
396, 38impbid 202 1 (𝜑 → (𝐴𝐵 ↔ (𝐴𝑚 𝐶) ⊆ (𝐵𝑚 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036  wcel 1987  wne 2790  wrex 2908  wss 3559  c0 3896  {csn 4153   × cxp 5077  wf 5848  (class class class)co 6610  𝑚 cmap 7809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-map 7811
This theorem is referenced by: (None)
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