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Theorem suppofss2d 7599
 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 6280 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
3 suppofssd.4 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
43ffnd 6280 . . . . . . 7 (𝜑𝐺 Fn 𝐴)
5 suppofssd.1 . . . . . . 7 (𝜑𝐴𝑉)
6 inidm 4048 . . . . . . 7 (𝐴𝐴) = 𝐴
7 eqidd 2827 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) = (𝐹𝑦))
8 eqidd 2827 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐺𝑦) = (𝐺𝑦))
92, 4, 5, 5, 6, 7, 8ofval 7167 . . . . . 6 ((𝜑𝑦𝐴) → ((𝐹𝑓 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
109adantr 474 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → ((𝐹𝑓 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
11 simpr 479 . . . . . 6 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → (𝐺𝑦) = 𝑍)
1211oveq2d 6922 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → ((𝐹𝑦)𝑋(𝐺𝑦)) = ((𝐹𝑦)𝑋𝑍))
13 suppofss2d.5 . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑥𝑋𝑍) = 𝑍)
1413ralrimiva 3176 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑥𝑋𝑍) = 𝑍)
1514adantr 474 . . . . . . 7 ((𝜑𝑦𝐴) → ∀𝑥𝐵 (𝑥𝑋𝑍) = 𝑍)
161ffvelrnda 6609 . . . . . . . 8 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐵)
17 simpr 479 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐹𝑦)) → 𝑥 = (𝐹𝑦))
1817oveq1d 6921 . . . . . . . . 9 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐹𝑦)) → (𝑥𝑋𝑍) = ((𝐹𝑦)𝑋𝑍))
1918eqeq1d 2828 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐹𝑦)) → ((𝑥𝑋𝑍) = 𝑍 ↔ ((𝐹𝑦)𝑋𝑍) = 𝑍))
2016, 19rspcdv 3530 . . . . . . 7 ((𝜑𝑦𝐴) → (∀𝑥𝐵 (𝑥𝑋𝑍) = 𝑍 → ((𝐹𝑦)𝑋𝑍) = 𝑍))
2115, 20mpd 15 . . . . . 6 ((𝜑𝑦𝐴) → ((𝐹𝑦)𝑋𝑍) = 𝑍)
2221adantr 474 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → ((𝐹𝑦)𝑋𝑍) = 𝑍)
2310, 12, 223eqtrd 2866 . . . 4 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → ((𝐹𝑓 𝑋𝐺)‘𝑦) = 𝑍)
2423ex 403 . . 3 ((𝜑𝑦𝐴) → ((𝐺𝑦) = 𝑍 → ((𝐹𝑓 𝑋𝐺)‘𝑦) = 𝑍))
2524ralrimiva 3176 . 2 (𝜑 → ∀𝑦𝐴 ((𝐺𝑦) = 𝑍 → ((𝐹𝑓 𝑋𝐺)‘𝑦) = 𝑍))
262, 4, 5, 5, 6offn 7169 . . 3 (𝜑 → (𝐹𝑓 𝑋𝐺) Fn 𝐴)
27 ssidd 3850 . . 3 (𝜑𝐴𝐴)
28 suppofssd.2 . . 3 (𝜑𝑍𝐵)
29 suppfnss 7585 . . 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 1502 . 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 386   = wceq 1658   ∈ wcel 2166  ∀wral 3118   ⊆ wss 3799   Fn wfn 6119  ⟶wf 6120  ‘cfv 6124  (class class class)co 6906   ∘𝑓 cof 7156   supp csupp 7560 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pr 5128  ax-un 7210 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-of 7158  df-supp 7561 This theorem is referenced by:  frlmphl  20488
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