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Theorem suppofss1d 7857
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 6508 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
3 suppofssd.4 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
43ffnd 6508 . . . . . . 7 (𝜑𝐺 Fn 𝐴)
5 suppofssd.1 . . . . . . 7 (𝜑𝐴𝑉)
6 inidm 4192 . . . . . . 7 (𝐴𝐴) = 𝐴
7 eqidd 2819 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) = (𝐹𝑦))
8 eqidd 2819 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐺𝑦) = (𝐺𝑦))
92, 4, 5, 5, 6, 7, 8ofval 7407 . . . . . 6 ((𝜑𝑦𝐴) → ((𝐹f 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
109adantr 481 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹f 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
11 simpr 485 . . . . . 6 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → (𝐹𝑦) = 𝑍)
1211oveq1d 7160 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹𝑦)𝑋(𝐺𝑦)) = (𝑍𝑋(𝐺𝑦)))
13 suppofss1d.5 . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑍𝑋𝑥) = 𝑍)
1413ralrimiva 3179 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍)
1514adantr 481 . . . . . . 7 ((𝜑𝑦𝐴) → ∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍)
163ffvelrnda 6843 . . . . . . . 8 ((𝜑𝑦𝐴) → (𝐺𝑦) ∈ 𝐵)
17 simpr 485 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → 𝑥 = (𝐺𝑦))
1817oveq2d 7161 . . . . . . . . 9 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → (𝑍𝑋𝑥) = (𝑍𝑋(𝐺𝑦)))
1918eqeq1d 2820 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → ((𝑍𝑋𝑥) = 𝑍 ↔ (𝑍𝑋(𝐺𝑦)) = 𝑍))
2016, 19rspcdv 3612 . . . . . . 7 ((𝜑𝑦𝐴) → (∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍 → (𝑍𝑋(𝐺𝑦)) = 𝑍))
2115, 20mpd 15 . . . . . 6 ((𝜑𝑦𝐴) → (𝑍𝑋(𝐺𝑦)) = 𝑍)
2221adantr 481 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → (𝑍𝑋(𝐺𝑦)) = 𝑍)
2310, 12, 223eqtrd 2857 . . . 4 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍)
2423ex 413 . . 3 ((𝜑𝑦𝐴) → ((𝐹𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍))
2524ralrimiva 3179 . 2 (𝜑 → ∀𝑦𝐴 ((𝐹𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍))
262, 4, 5, 5, 6offn 7409 . . 3 (𝜑 → (𝐹f 𝑋𝐺) Fn 𝐴)
27 ssidd 3987 . . 3 (𝜑𝐴𝐴)
28 suppofssd.2 . . 3 (𝜑𝑍𝐵)
29 suppfnss 7844 . . 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 1369 . 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 396   = wceq 1528  wcel 2105  wral 3135  wss 3933   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  f cof 7396   supp csupp 7819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  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 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-supp 7820
This theorem is referenced by:  frlmphllem  20852  rrxcph  23922
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