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Theorem extmptsuppeq 7264
Description: The support of an extended function is the same as the original. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 30-Jun-2019.)
Hypotheses
Ref Expression
extmptsuppeq.b (𝜑𝐵𝑊)
extmptsuppeq.a (𝜑𝐴𝐵)
extmptsuppeq.z ((𝜑𝑛 ∈ (𝐵𝐴)) → 𝑋 = 𝑍)
Assertion
Ref Expression
extmptsuppeq (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝑛,𝑍   𝜑,𝑛
Allowed substitution hints:   𝑊(𝑛)   𝑋(𝑛)

Proof of Theorem extmptsuppeq
StepHypRef Expression
1 extmptsuppeq.a . . . . . . . . 9 (𝜑𝐴𝐵)
21adantl 482 . . . . . . . 8 ((𝑍 ∈ V ∧ 𝜑) → 𝐴𝐵)
32sseld 3582 . . . . . . 7 ((𝑍 ∈ V ∧ 𝜑) → (𝑛𝐴𝑛𝐵))
43anim1d 587 . . . . . 6 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋 ∈ (V ∖ {𝑍})) → (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))))
5 eldif 3565 . . . . . . . . . . . . 13 (𝑛 ∈ (𝐵𝐴) ↔ (𝑛𝐵 ∧ ¬ 𝑛𝐴))
6 extmptsuppeq.z . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝐵𝐴)) → 𝑋 = 𝑍)
76adantll 749 . . . . . . . . . . . . 13 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ (𝐵𝐴)) → 𝑋 = 𝑍)
85, 7sylan2br 493 . . . . . . . . . . . 12 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵 ∧ ¬ 𝑛𝐴)) → 𝑋 = 𝑍)
98expr 642 . . . . . . . . . . 11 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛𝐵) → (¬ 𝑛𝐴𝑋 = 𝑍))
10 elsn2g 4181 . . . . . . . . . . . . 13 (𝑍 ∈ V → (𝑋 ∈ {𝑍} ↔ 𝑋 = 𝑍))
11 elndif 3712 . . . . . . . . . . . . 13 (𝑋 ∈ {𝑍} → ¬ 𝑋 ∈ (V ∖ {𝑍}))
1210, 11syl6bir 244 . . . . . . . . . . . 12 (𝑍 ∈ V → (𝑋 = 𝑍 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
1312ad2antrr 761 . . . . . . . . . . 11 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛𝐵) → (𝑋 = 𝑍 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
149, 13syld 47 . . . . . . . . . 10 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛𝐵) → (¬ 𝑛𝐴 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
1514con4d 114 . . . . . . . . 9 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛𝐵) → (𝑋 ∈ (V ∖ {𝑍}) → 𝑛𝐴))
1615impr 648 . . . . . . . 8 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))) → 𝑛𝐴)
17 simprr 795 . . . . . . . 8 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))) → 𝑋 ∈ (V ∖ {𝑍}))
1816, 17jca 554 . . . . . . 7 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))) → (𝑛𝐴𝑋 ∈ (V ∖ {𝑍})))
1918ex 450 . . . . . 6 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐵𝑋 ∈ (V ∖ {𝑍})) → (𝑛𝐴𝑋 ∈ (V ∖ {𝑍}))))
204, 19impbid 202 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋 ∈ (V ∖ {𝑍})) ↔ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))))
2120rabbidva2 3174 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → {𝑛𝐴𝑋 ∈ (V ∖ {𝑍})} = {𝑛𝐵𝑋 ∈ (V ∖ {𝑍})})
22 eqid 2621 . . . . 5 (𝑛𝐴𝑋) = (𝑛𝐴𝑋)
23 extmptsuppeq.b . . . . . . 7 (𝜑𝐵𝑊)
2423, 1ssexd 4765 . . . . . 6 (𝜑𝐴 ∈ V)
2524adantl 482 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ∈ V)
26 simpl 473 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
2722, 25, 26mptsuppdifd 7262 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋) supp 𝑍) = {𝑛𝐴𝑋 ∈ (V ∖ {𝑍})})
28 eqid 2621 . . . . 5 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
2923adantl 482 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝐵𝑊)
3028, 29, 26mptsuppdifd 7262 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐵𝑋) supp 𝑍) = {𝑛𝐵𝑋 ∈ (V ∖ {𝑍})})
3121, 27, 303eqtr4d 2665 . . 3 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
3231ex 450 . 2 (𝑍 ∈ V → (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍)))
33 simpr 477 . . . . . 6 (((𝑛𝐴𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
3433con3i 150 . . . . 5 𝑍 ∈ V → ¬ ((𝑛𝐴𝑋) ∈ V ∧ 𝑍 ∈ V))
35 supp0prc 7243 . . . . 5 (¬ ((𝑛𝐴𝑋) ∈ V ∧ 𝑍 ∈ V) → ((𝑛𝐴𝑋) supp 𝑍) = ∅)
3634, 35syl 17 . . . 4 𝑍 ∈ V → ((𝑛𝐴𝑋) supp 𝑍) = ∅)
37 simpr 477 . . . . . 6 (((𝑛𝐵𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
3837con3i 150 . . . . 5 𝑍 ∈ V → ¬ ((𝑛𝐵𝑋) ∈ V ∧ 𝑍 ∈ V))
39 supp0prc 7243 . . . . 5 (¬ ((𝑛𝐵𝑋) ∈ V ∧ 𝑍 ∈ V) → ((𝑛𝐵𝑋) supp 𝑍) = ∅)
4038, 39syl 17 . . . 4 𝑍 ∈ V → ((𝑛𝐵𝑋) supp 𝑍) = ∅)
4136, 40eqtr4d 2658 . . 3 𝑍 ∈ V → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
4241a1d 25 . 2 𝑍 ∈ V → (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍)))
4332, 42pm2.61i 176 1 (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  cdif 3552  wss 3555  c0 3891  {csn 4148  cmpt 4673  (class class class)co 6604   supp csupp 7240
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-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-supp 7241
This theorem is referenced by:  cantnfrescl  8517  cantnfres  8518
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