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Theorem funsssuppss 7852
 Description: The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.)
Assertion
Ref Expression
funsssuppss ((Fun 𝐺𝐹𝐺𝐺𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))

Proof of Theorem funsssuppss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funss 6362 . . . . . . . . . 10 (𝐹𝐺 → (Fun 𝐺 → Fun 𝐹))
21impcom 411 . . . . . . . . 9 ((Fun 𝐺𝐹𝐺) → Fun 𝐹)
32funfnd 6374 . . . . . . . 8 ((Fun 𝐺𝐹𝐺) → 𝐹 Fn dom 𝐹)
4 funfn 6373 . . . . . . . . . 10 (Fun 𝐺𝐺 Fn dom 𝐺)
54biimpi 219 . . . . . . . . 9 (Fun 𝐺𝐺 Fn dom 𝐺)
65adantr 484 . . . . . . . 8 ((Fun 𝐺𝐹𝐺) → 𝐺 Fn dom 𝐺)
73, 6jca 515 . . . . . . 7 ((Fun 𝐺𝐹𝐺) → (𝐹 Fn dom 𝐹𝐺 Fn dom 𝐺))
873adant3 1129 . . . . . 6 ((Fun 𝐺𝐹𝐺𝐺𝑉) → (𝐹 Fn dom 𝐹𝐺 Fn dom 𝐺))
98adantr 484 . . . . 5 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → (𝐹 Fn dom 𝐹𝐺 Fn dom 𝐺))
10 dmss 5758 . . . . . . . 8 (𝐹𝐺 → dom 𝐹 ⊆ dom 𝐺)
11103ad2ant2 1131 . . . . . . 7 ((Fun 𝐺𝐹𝐺𝐺𝑉) → dom 𝐹 ⊆ dom 𝐺)
1211adantr 484 . . . . . 6 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → dom 𝐹 ⊆ dom 𝐺)
13 dmexg 7608 . . . . . . . 8 (𝐺𝑉 → dom 𝐺 ∈ V)
14133ad2ant3 1132 . . . . . . 7 ((Fun 𝐺𝐹𝐺𝐺𝑉) → dom 𝐺 ∈ V)
1514adantr 484 . . . . . 6 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → dom 𝐺 ∈ V)
16 simpr 488 . . . . . 6 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
1712, 15, 163jca 1125 . . . . 5 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → (dom 𝐹 ⊆ dom 𝐺 ∧ dom 𝐺 ∈ V ∧ 𝑍 ∈ V))
189, 17jca 515 . . . 4 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → ((𝐹 Fn dom 𝐹𝐺 Fn dom 𝐺) ∧ (dom 𝐹 ⊆ dom 𝐺 ∧ dom 𝐺 ∈ V ∧ 𝑍 ∈ V)))
19 funssfv 6682 . . . . . . . . 9 ((Fun 𝐺𝐹𝐺𝑥 ∈ dom 𝐹) → (𝐺𝑥) = (𝐹𝑥))
20193expa 1115 . . . . . . . 8 (((Fun 𝐺𝐹𝐺) ∧ 𝑥 ∈ dom 𝐹) → (𝐺𝑥) = (𝐹𝑥))
21 eqeq1 2828 . . . . . . . . 9 ((𝐺𝑥) = (𝐹𝑥) → ((𝐺𝑥) = 𝑍 ↔ (𝐹𝑥) = 𝑍))
2221biimpd 232 . . . . . . . 8 ((𝐺𝑥) = (𝐹𝑥) → ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍))
2320, 22syl 17 . . . . . . 7 (((Fun 𝐺𝐹𝐺) ∧ 𝑥 ∈ dom 𝐹) → ((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍))
2423ralrimiva 3177 . . . . . 6 ((Fun 𝐺𝐹𝐺) → ∀𝑥 ∈ dom 𝐹((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍))
25243adant3 1129 . . . . 5 ((Fun 𝐺𝐹𝐺𝐺𝑉) → ∀𝑥 ∈ dom 𝐹((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍))
2625adantr 484 . . . 4 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → ∀𝑥 ∈ dom 𝐹((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍))
27 suppfnss 7851 . . . 4 (((𝐹 Fn dom 𝐹𝐺 Fn dom 𝐺) ∧ (dom 𝐹 ⊆ dom 𝐺 ∧ dom 𝐺 ∈ V ∧ 𝑍 ∈ V)) → (∀𝑥 ∈ dom 𝐹((𝐺𝑥) = 𝑍 → (𝐹𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
2818, 26, 27sylc 65 . . 3 (((Fun 𝐺𝐹𝐺𝐺𝑉) ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
2928expcom 417 . 2 (𝑍 ∈ V → ((Fun 𝐺𝐹𝐺𝐺𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
30 ssid 3975 . . . 4 ∅ ⊆ ∅
31 simpr 488 . . . . . 6 ((𝐹 ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
32 supp0prc 7829 . . . . . 6 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
3331, 32nsyl5 162 . . . . 5 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
34 simpr 488 . . . . . 6 ((𝐺 ∈ V ∧ 𝑍 ∈ V) → 𝑍 ∈ V)
35 supp0prc 7829 . . . . . 6 (¬ (𝐺 ∈ V ∧ 𝑍 ∈ V) → (𝐺 supp 𝑍) = ∅)
3634, 35nsyl5 162 . . . . 5 𝑍 ∈ V → (𝐺 supp 𝑍) = ∅)
3733, 36sseq12d 3986 . . . 4 𝑍 ∈ V → ((𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍) ↔ ∅ ⊆ ∅))
3830, 37mpbiri 261 . . 3 𝑍 ∈ V → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
3938a1d 25 . 2 𝑍 ∈ V → ((Fun 𝐺𝐹𝐺𝐺𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
4029, 39pm2.61i 185 1 ((Fun 𝐺𝐹𝐺𝐺𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3133  Vcvv 3480   ⊆ wss 3919  ∅c0 4276  dom cdm 5542  Fun wfun 6337   Fn wfn 6338  ‘cfv 6343  (class class class)co 7149   supp csupp 7826 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-supp 7827 This theorem is referenced by:  tdeglem4  24664
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