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Theorem suppofss2d 7861
 Description: Condition for the support of a function operation to be a subset of the support of the right function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
Hypotheses
Ref Expression
suppofssd.1 (𝜑𝐴𝑉)
suppofssd.2 (𝜑𝑍𝐵)
suppofssd.3 (𝜑𝐹:𝐴𝐵)
suppofssd.4 (𝜑𝐺:𝐴𝐵)
suppofss2d.5 ((𝜑𝑥𝐵) → (𝑥𝑋𝑍) = 𝑍)
Assertion
Ref Expression
suppofss2d (𝜑 → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺   𝑥,𝑋   𝑥,𝑍   𝜑,𝑥
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem suppofss2d
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 suppofssd.3 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
21ffnd 6508 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
3 suppofssd.4 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
43ffnd 6508 . . . . . . 7 (𝜑𝐺 Fn 𝐴)
5 suppofssd.1 . . . . . . 7 (𝜑𝐴𝑉)
6 inidm 4193 . . . . . . 7 (𝐴𝐴) = 𝐴
7 eqidd 2820 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) = (𝐹𝑦))
8 eqidd 2820 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐺𝑦) = (𝐺𝑦))
92, 4, 5, 5, 6, 7, 8ofval 7410 . . . . . 6 ((𝜑𝑦𝐴) → ((𝐹f 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
109adantr 483 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → ((𝐹f 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
11 simpr 487 . . . . . 6 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → (𝐺𝑦) = 𝑍)
1211oveq2d 7164 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → ((𝐹𝑦)𝑋(𝐺𝑦)) = ((𝐹𝑦)𝑋𝑍))
13 suppofss2d.5 . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑥𝑋𝑍) = 𝑍)
1413ralrimiva 3180 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑥𝑋𝑍) = 𝑍)
1514adantr 483 . . . . . . 7 ((𝜑𝑦𝐴) → ∀𝑥𝐵 (𝑥𝑋𝑍) = 𝑍)
161ffvelrnda 6844 . . . . . . . 8 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐵)
17 simpr 487 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐹𝑦)) → 𝑥 = (𝐹𝑦))
1817oveq1d 7163 . . . . . . . . 9 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐹𝑦)) → (𝑥𝑋𝑍) = ((𝐹𝑦)𝑋𝑍))
1918eqeq1d 2821 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐹𝑦)) → ((𝑥𝑋𝑍) = 𝑍 ↔ ((𝐹𝑦)𝑋𝑍) = 𝑍))
2016, 19rspcdv 3613 . . . . . . 7 ((𝜑𝑦𝐴) → (∀𝑥𝐵 (𝑥𝑋𝑍) = 𝑍 → ((𝐹𝑦)𝑋𝑍) = 𝑍))
2115, 20mpd 15 . . . . . 6 ((𝜑𝑦𝐴) → ((𝐹𝑦)𝑋𝑍) = 𝑍)
2221adantr 483 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → ((𝐹𝑦)𝑋𝑍) = 𝑍)
2310, 12, 223eqtrd 2858 . . . 4 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍)
2423ex 415 . . 3 ((𝜑𝑦𝐴) → ((𝐺𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍))
2524ralrimiva 3180 . 2 (𝜑 → ∀𝑦𝐴 ((𝐺𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍))
262, 4, 5, 5, 6offn 7412 . . 3 (𝜑 → (𝐹f 𝑋𝐺) Fn 𝐴)
27 ssidd 3988 . . 3 (𝜑𝐴𝐴)
28 suppofssd.2 . . 3 (𝜑𝑍𝐵)
29 suppfnss 7847 . . 3 ((((𝐹f 𝑋𝐺) Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐴𝐴𝐴𝑉𝑍𝐵)) → (∀𝑦𝐴 ((𝐺𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍) → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍)))
3026, 4, 27, 5, 28, 29syl23anc 1371 . 2 (𝜑 → (∀𝑦𝐴 ((𝐺𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍) → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍)))
3125, 30mpd 15 1 (𝜑 → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1530   ∈ wcel 2107  ∀wral 3136   ⊆ wss 3934   Fn wfn 6343  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148   ∘f cof 7399   supp csupp 7822 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 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-supp 7823 This theorem is referenced by:  frlmphl  20917
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