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Theorem suppofss2d 8212
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 6721 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
3 suppofssd.4 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
43ffnd 6721 . . . . . . 7 (𝜑𝐺 Fn 𝐴)
5 suppofssd.1 . . . . . . 7 (𝜑𝐴𝑉)
6 inidm 4217 . . . . . . 7 (𝐴𝐴) = 𝐴
7 eqidd 2727 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) = (𝐹𝑦))
8 eqidd 2727 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐺𝑦) = (𝐺𝑦))
92, 4, 5, 5, 6, 7, 8ofval 7693 . . . . . 6 ((𝜑𝑦𝐴) → ((𝐹f 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
109adantr 479 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → ((𝐹f 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
11 simpr 483 . . . . . 6 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → (𝐺𝑦) = 𝑍)
1211oveq2d 7432 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → ((𝐹𝑦)𝑋(𝐺𝑦)) = ((𝐹𝑦)𝑋𝑍))
13 suppofss2d.5 . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑥𝑋𝑍) = 𝑍)
1413ralrimiva 3136 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑥𝑋𝑍) = 𝑍)
1514adantr 479 . . . . . . 7 ((𝜑𝑦𝐴) → ∀𝑥𝐵 (𝑥𝑋𝑍) = 𝑍)
161ffvelcdmda 7090 . . . . . . . 8 ((𝜑𝑦𝐴) → (𝐹𝑦) ∈ 𝐵)
17 simpr 483 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐹𝑦)) → 𝑥 = (𝐹𝑦))
1817oveq1d 7431 . . . . . . . . 9 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐹𝑦)) → (𝑥𝑋𝑍) = ((𝐹𝑦)𝑋𝑍))
1918eqeq1d 2728 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐹𝑦)) → ((𝑥𝑋𝑍) = 𝑍 ↔ ((𝐹𝑦)𝑋𝑍) = 𝑍))
2016, 19rspcdv 3599 . . . . . . 7 ((𝜑𝑦𝐴) → (∀𝑥𝐵 (𝑥𝑋𝑍) = 𝑍 → ((𝐹𝑦)𝑋𝑍) = 𝑍))
2115, 20mpd 15 . . . . . 6 ((𝜑𝑦𝐴) → ((𝐹𝑦)𝑋𝑍) = 𝑍)
2221adantr 479 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → ((𝐹𝑦)𝑋𝑍) = 𝑍)
2310, 12, 223eqtrd 2770 . . . 4 (((𝜑𝑦𝐴) ∧ (𝐺𝑦) = 𝑍) → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍)
2423ex 411 . . 3 ((𝜑𝑦𝐴) → ((𝐺𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍))
2524ralrimiva 3136 . 2 (𝜑 → ∀𝑦𝐴 ((𝐺𝑦) = 𝑍 → ((𝐹f 𝑋𝐺)‘𝑦) = 𝑍))
262, 4, 5, 5, 6offn 7695 . . 3 (𝜑 → (𝐹f 𝑋𝐺) Fn 𝐴)
27 ssidd 4002 . . 3 (𝜑𝐴𝐴)
28 suppofssd.2 . . 3 (𝜑𝑍𝐵)
29 suppfnss 8195 . . 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 1374 . 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 394   = wceq 1534  wcel 2099  wral 3051  wss 3946   Fn wfn 6541  wf 6542  cfv 6546  (class class class)co 7416  f cof 7680   supp csupp 8166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-supp 8167
This theorem is referenced by:  frlmphl  21775
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