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Theorem suppofss1d 7503
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)
2 ffn 6207 . . . . . . . 8 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
31, 2syl 17 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
4 suppofssd.4 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
5 ffn 6207 . . . . . . . 8 (𝐺:𝐴𝐵𝐺 Fn 𝐴)
64, 5syl 17 . . . . . . 7 (𝜑𝐺 Fn 𝐴)
7 suppofssd.1 . . . . . . 7 (𝜑𝐴𝑉)
8 inidm 3966 . . . . . . 7 (𝐴𝐴) = 𝐴
9 eqidd 2762 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) = (𝐹𝑦))
10 eqidd 2762 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐺𝑦) = (𝐺𝑦))
113, 6, 7, 7, 8, 9, 10ofval 7073 . . . . . 6 ((𝜑𝑦𝐴) → ((𝐹𝑓 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
1211adantr 472 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹𝑓 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
13 simpr 479 . . . . . 6 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → (𝐹𝑦) = 𝑍)
1413oveq1d 6830 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹𝑦)𝑋(𝐺𝑦)) = (𝑍𝑋(𝐺𝑦)))
15 suppofss1d.5 . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑍𝑋𝑥) = 𝑍)
1615ralrimiva 3105 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍)
1716adantr 472 . . . . . . 7 ((𝜑𝑦𝐴) → ∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍)
184ffvelrnda 6524 . . . . . . . 8 ((𝜑𝑦𝐴) → (𝐺𝑦) ∈ 𝐵)
19 simpr 479 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → 𝑥 = (𝐺𝑦))
2019oveq2d 6831 . . . . . . . . 9 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → (𝑍𝑋𝑥) = (𝑍𝑋(𝐺𝑦)))
2120eqeq1d 2763 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → ((𝑍𝑋𝑥) = 𝑍 ↔ (𝑍𝑋(𝐺𝑦)) = 𝑍))
2218, 21rspcdv 3453 . . . . . . 7 ((𝜑𝑦𝐴) → (∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍 → (𝑍𝑋(𝐺𝑦)) = 𝑍))
2317, 22mpd 15 . . . . . 6 ((𝜑𝑦𝐴) → (𝑍𝑋(𝐺𝑦)) = 𝑍)
2423adantr 472 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → (𝑍𝑋(𝐺𝑦)) = 𝑍)
2512, 14, 243eqtrd 2799 . . . 4 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹𝑓 𝑋𝐺)‘𝑦) = 𝑍)
2625ex 449 . . 3 ((𝜑𝑦𝐴) → ((𝐹𝑦) = 𝑍 → ((𝐹𝑓 𝑋𝐺)‘𝑦) = 𝑍))
2726ralrimiva 3105 . 2 (𝜑 → ∀𝑦𝐴 ((𝐹𝑦) = 𝑍 → ((𝐹𝑓 𝑋𝐺)‘𝑦) = 𝑍))
283, 6, 7, 7, 8offn 7075 . . 3 (𝜑 → (𝐹𝑓 𝑋𝐺) Fn 𝐴)
29 ssid 3766 . . . 4 𝐴𝐴
3029a1i 11 . . 3 (𝜑𝐴𝐴)
31 suppofssd.2 . . 3 (𝜑𝑍𝐵)
32 suppfnss 7490 . . 3 ((((𝐹𝑓 𝑋𝐺) Fn 𝐴𝐹 Fn 𝐴) ∧ (𝐴𝐴𝐴𝑉𝑍𝐵)) → (∀𝑦𝐴 ((𝐹𝑦) = 𝑍 → ((𝐹𝑓 𝑋𝐺)‘𝑦) = 𝑍) → ((𝐹𝑓 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍)))
3328, 3, 30, 7, 31, 32syl23anc 1484 . 2 (𝜑 → (∀𝑦𝐴 ((𝐹𝑦) = 𝑍 → ((𝐹𝑓 𝑋𝐺)‘𝑦) = 𝑍) → ((𝐹𝑓 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍)))
3427, 33mpd 15 1 (𝜑 → ((𝐹𝑓 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2140  wral 3051  wss 3716   Fn wfn 6045  wf 6046  cfv 6050  (class class class)co 6815  𝑓 cof 7062   supp csupp 7465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-of 7064  df-supp 7466
This theorem is referenced by:  frlmphllem  20342  rrxcph  23401
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