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Theorem suppofss1d 7864
Description: Condition for the support of a function operation to be a subset of the support of the left 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)
suppofss1d.5 ((𝜑𝑥𝐵) → (𝑍𝑋𝑥) = 𝑍)
Assertion
Ref Expression
suppofss1d (𝜑 → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ (𝐹 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 suppofss1d
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 suppofssd.3 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
21ffnd 6504 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
3 suppofssd.4 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
43ffnd 6504 . . . . . . 7 (𝜑𝐺 Fn 𝐴)
5 suppofssd.1 . . . . . . 7 (𝜑𝐴𝑉)
6 inidm 4180 . . . . . . 7 (𝐴𝐴) = 𝐴
7 eqidd 2825 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) = (𝐹𝑦))
8 eqidd 2825 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐺𝑦) = (𝐺𝑦))
92, 4, 5, 5, 6, 7, 8ofval 7412 . . . . . 6 ((𝜑𝑦𝐴) → ((𝐹f 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
109adantr 484 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹f 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
11 simpr 488 . . . . . 6 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → (𝐹𝑦) = 𝑍)
1211oveq1d 7164 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹𝑦)𝑋(𝐺𝑦)) = (𝑍𝑋(𝐺𝑦)))
13 suppofss1d.5 . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑍𝑋𝑥) = 𝑍)
1413ralrimiva 3177 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍)
1514adantr 484 . . . . . . 7 ((𝜑𝑦𝐴) → ∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍)
163ffvelrnda 6842 . . . . . . . 8 ((𝜑𝑦𝐴) → (𝐺𝑦) ∈ 𝐵)
17 simpr 488 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → 𝑥 = (𝐺𝑦))
1817oveq2d 7165 . . . . . . . . 9 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → (𝑍𝑋𝑥) = (𝑍𝑋(𝐺𝑦)))
1918eqeq1d 2826 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → ((𝑍𝑋𝑥) = 𝑍 ↔ (𝑍𝑋(𝐺𝑦)) = 𝑍))
2016, 19rspcdv 3601 . . . . . . 7 ((𝜑𝑦𝐴) → (∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍 → (𝑍𝑋(𝐺𝑦)) = 𝑍))
2115, 20mpd 15 . . . . . 6 ((𝜑𝑦𝐴) → (𝑍𝑋(𝐺𝑦)) = 𝑍)
2221adantr 484 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → (𝑍𝑋(𝐺𝑦)) = 𝑍)
2310, 12, 223eqtrd 2863 . . . 4 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍)
2423ex 416 . . 3 ((𝜑𝑦𝐴) → ((𝐹𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍))
2524ralrimiva 3177 . 2 (𝜑 → ∀𝑦𝐴 ((𝐹𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍))
262, 4, 5, 5, 6offn 7414 . . 3 (𝜑 → (𝐹f 𝑋𝐺) Fn 𝐴)
27 ssidd 3976 . . 3 (𝜑𝐴𝐴)
28 suppofssd.2 . . 3 (𝜑𝑍𝐵)
29 suppfnss 7851 . . 3 ((((𝐹f 𝑋𝐺) Fn 𝐴𝐹 Fn 𝐴) ∧ (𝐴𝐴𝐴𝑉𝑍𝐵)) → (∀𝑦𝐴 ((𝐹𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍) → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍)))
3026, 2, 27, 5, 28, 29syl23anc 1374 . 2 (𝜑 → (∀𝑦𝐴 ((𝐹𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍) → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍)))
3125, 30mpd 15 1 (𝜑 → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  wss 3919   Fn wfn 6338  wf 6339  cfv 6343  (class class class)co 7149  f cof 7401   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-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-of 7403  df-supp 7827
This theorem is referenced by:  frlmphllem  20924  rrxcph  23999
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