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Theorem suppofss1d 8190
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 6712 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
3 suppofssd.4 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
43ffnd 6712 . . . . . . 7 (𝜑𝐺 Fn 𝐴)
5 suppofssd.1 . . . . . . 7 (𝜑𝐴𝑉)
6 inidm 4213 . . . . . . 7 (𝐴𝐴) = 𝐴
7 eqidd 2727 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) = (𝐹𝑦))
8 eqidd 2727 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐺𝑦) = (𝐺𝑦))
92, 4, 5, 5, 6, 7, 8ofval 7678 . . . . . 6 ((𝜑𝑦𝐴) → ((𝐹f 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
109adantr 480 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹f 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
11 simpr 484 . . . . . 6 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → (𝐹𝑦) = 𝑍)
1211oveq1d 7420 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹𝑦)𝑋(𝐺𝑦)) = (𝑍𝑋(𝐺𝑦)))
13 suppofss1d.5 . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑍𝑋𝑥) = 𝑍)
1413ralrimiva 3140 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍)
1514adantr 480 . . . . . . 7 ((𝜑𝑦𝐴) → ∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍)
163ffvelcdmda 7080 . . . . . . . 8 ((𝜑𝑦𝐴) → (𝐺𝑦) ∈ 𝐵)
17 simpr 484 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → 𝑥 = (𝐺𝑦))
1817oveq2d 7421 . . . . . . . . 9 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → (𝑍𝑋𝑥) = (𝑍𝑋(𝐺𝑦)))
1918eqeq1d 2728 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → ((𝑍𝑋𝑥) = 𝑍 ↔ (𝑍𝑋(𝐺𝑦)) = 𝑍))
2016, 19rspcdv 3598 . . . . . . 7 ((𝜑𝑦𝐴) → (∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍 → (𝑍𝑋(𝐺𝑦)) = 𝑍))
2115, 20mpd 15 . . . . . 6 ((𝜑𝑦𝐴) → (𝑍𝑋(𝐺𝑦)) = 𝑍)
2221adantr 480 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → (𝑍𝑋(𝐺𝑦)) = 𝑍)
2310, 12, 223eqtrd 2770 . . . 4 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍)
2423ex 412 . . 3 ((𝜑𝑦𝐴) → ((𝐹𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍))
2524ralrimiva 3140 . 2 (𝜑 → ∀𝑦𝐴 ((𝐹𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍))
262, 4, 5, 5, 6offn 7680 . . 3 (𝜑 → (𝐹f 𝑋𝐺) Fn 𝐴)
27 ssidd 4000 . . 3 (𝜑𝐴𝐴)
28 suppofssd.2 . . 3 (𝜑𝑍𝐵)
29 suppfnss 8174 . . 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 395   = wceq 1533  wcel 2098  wral 3055  wss 3943   Fn wfn 6532  wf 6533  cfv 6537  (class class class)co 7405  f cof 7665   supp csupp 8146
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7667  df-supp 8147
This theorem is referenced by:  frlmphllem  21675  rrxcph  25275
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