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Theorem extmptsuppeq 7853
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 484 . . . . . . . 8 ((𝑍 ∈ V ∧ 𝜑) → 𝐴𝐵)
32sseld 3965 . . . . . . 7 ((𝑍 ∈ V ∧ 𝜑) → (𝑛𝐴𝑛𝐵))
43anim1d 612 . . . . . 6 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋 ∈ (V ∖ {𝑍})) → (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))))
5 eldif 3945 . . . . . . . . . . . . 13 (𝑛 ∈ (𝐵𝐴) ↔ (𝑛𝐵 ∧ ¬ 𝑛𝐴))
6 extmptsuppeq.z . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (𝐵𝐴)) → 𝑋 = 𝑍)
76adantll 712 . . . . . . . . . . . . 13 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛 ∈ (𝐵𝐴)) → 𝑋 = 𝑍)
85, 7sylan2br 596 . . . . . . . . . . . 12 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵 ∧ ¬ 𝑛𝐴)) → 𝑋 = 𝑍)
98expr 459 . . . . . . . . . . 11 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛𝐵) → (¬ 𝑛𝐴𝑋 = 𝑍))
10 elsn2g 4602 . . . . . . . . . . . . 13 (𝑍 ∈ V → (𝑋 ∈ {𝑍} ↔ 𝑋 = 𝑍))
11 elndif 4104 . . . . . . . . . . . . 13 (𝑋 ∈ {𝑍} → ¬ 𝑋 ∈ (V ∖ {𝑍}))
1210, 11syl6bir 256 . . . . . . . . . . . 12 (𝑍 ∈ V → (𝑋 = 𝑍 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
1312ad2antrr 724 . . . . . . . . . . 11 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛𝐵) → (𝑋 = 𝑍 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
149, 13syld 47 . . . . . . . . . 10 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛𝐵) → (¬ 𝑛𝐴 → ¬ 𝑋 ∈ (V ∖ {𝑍})))
1514con4d 115 . . . . . . . . 9 (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑛𝐵) → (𝑋 ∈ (V ∖ {𝑍}) → 𝑛𝐴))
1615impr 457 . . . . . . . 8 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))) → 𝑛𝐴)
17 simprr 771 . . . . . . . 8 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))) → 𝑋 ∈ (V ∖ {𝑍}))
1816, 17jca 514 . . . . . . 7 (((𝑍 ∈ V ∧ 𝜑) ∧ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))) → (𝑛𝐴𝑋 ∈ (V ∖ {𝑍})))
1918ex 415 . . . . . 6 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐵𝑋 ∈ (V ∖ {𝑍})) → (𝑛𝐴𝑋 ∈ (V ∖ {𝑍}))))
204, 19impbid 214 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋 ∈ (V ∖ {𝑍})) ↔ (𝑛𝐵𝑋 ∈ (V ∖ {𝑍}))))
2120rabbidva2 3476 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → {𝑛𝐴𝑋 ∈ (V ∖ {𝑍})} = {𝑛𝐵𝑋 ∈ (V ∖ {𝑍})})
22 eqid 2821 . . . . 5 (𝑛𝐴𝑋) = (𝑛𝐴𝑋)
23 extmptsuppeq.b . . . . . . 7 (𝜑𝐵𝑊)
2423, 1ssexd 5227 . . . . . 6 (𝜑𝐴 ∈ V)
2524adantl 484 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝐴 ∈ V)
26 simpl 485 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
2722, 25, 26mptsuppdifd 7851 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋) supp 𝑍) = {𝑛𝐴𝑋 ∈ (V ∖ {𝑍})})
28 eqid 2821 . . . . 5 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
2923adantl 484 . . . . 5 ((𝑍 ∈ V ∧ 𝜑) → 𝐵𝑊)
3028, 29, 26mptsuppdifd 7851 . . . 4 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐵𝑋) supp 𝑍) = {𝑛𝐵𝑋 ∈ (V ∖ {𝑍})})
3121, 27, 303eqtr4d 2866 . . 3 ((𝑍 ∈ V ∧ 𝜑) → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
3231ex 415 . 2 (𝑍 ∈ V → (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍)))
33 simpr 487 . . . . . 6 (((𝑛𝐴𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
3433con3i 157 . . . . 5 𝑍 ∈ V → ¬ ((𝑛𝐴𝑋) ∈ V ∧ 𝑍 ∈ V))
35 supp0prc 7832 . . . . 5 (¬ ((𝑛𝐴𝑋) ∈ V ∧ 𝑍 ∈ V) → ((𝑛𝐴𝑋) supp 𝑍) = ∅)
3634, 35syl 17 . . . 4 𝑍 ∈ V → ((𝑛𝐴𝑋) supp 𝑍) = ∅)
37 simpr 487 . . . . . 6 (((𝑛𝐵𝑋) ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
3837con3i 157 . . . . 5 𝑍 ∈ V → ¬ ((𝑛𝐵𝑋) ∈ V ∧ 𝑍 ∈ V))
39 supp0prc 7832 . . . . 5 (¬ ((𝑛𝐵𝑋) ∈ V ∧ 𝑍 ∈ V) → ((𝑛𝐵𝑋) supp 𝑍) = ∅)
4038, 39syl 17 . . . 4 𝑍 ∈ V → ((𝑛𝐵𝑋) supp 𝑍) = ∅)
4136, 40eqtr4d 2859 . . 3 𝑍 ∈ V → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
4241a1d 25 . 2 𝑍 ∈ V → (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍)))
4332, 42pm2.61i 184 1 (𝜑 → ((𝑛𝐴𝑋) supp 𝑍) = ((𝑛𝐵𝑋) supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1533  wcel 2110  {crab 3142  Vcvv 3494  cdif 3932  wss 3935  c0 4290  {csn 4566  cmpt 5145  (class class class)co 7155   supp csupp 7829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  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 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-supp 7830
This theorem is referenced by:  cantnfrescl  9138  cantnfres  9139
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