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Theorem suppofss1d 7219
 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 5958 . . . . . . . 8 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
31, 2syl 17 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
4 suppofssd.4 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
5 ffn 5958 . . . . . . . 8 (𝐺:𝐴𝐵𝐺 Fn 𝐴)
64, 5syl 17 . . . . . . 7 (𝜑𝐺 Fn 𝐴)
7 suppofssd.1 . . . . . . 7 (𝜑𝐴𝑉)
8 inidm 3784 . . . . . . 7 (𝐴𝐴) = 𝐴
9 eqidd 2611 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐹𝑦) = (𝐹𝑦))
10 eqidd 2611 . . . . . . 7 ((𝜑𝑦𝐴) → (𝐺𝑦) = (𝐺𝑦))
113, 6, 7, 7, 8, 9, 10ofval 6804 . . . . . 6 ((𝜑𝑦𝐴) → ((𝐹𝑓 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
1211adantr 480 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹𝑓 𝑋𝐺)‘𝑦) = ((𝐹𝑦)𝑋(𝐺𝑦)))
13 simpr 476 . . . . . 6 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → (𝐹𝑦) = 𝑍)
1413oveq1d 6564 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹𝑦)𝑋(𝐺𝑦)) = (𝑍𝑋(𝐺𝑦)))
15 suppofss1d.5 . . . . . . . . 9 ((𝜑𝑥𝐵) → (𝑍𝑋𝑥) = 𝑍)
1615ralrimiva 2949 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍)
1716adantr 480 . . . . . . 7 ((𝜑𝑦𝐴) → ∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍)
184ffvelrnda 6267 . . . . . . . 8 ((𝜑𝑦𝐴) → (𝐺𝑦) ∈ 𝐵)
19 simpr 476 . . . . . . . . . 10 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → 𝑥 = (𝐺𝑦))
2019oveq2d 6565 . . . . . . . . 9 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → (𝑍𝑋𝑥) = (𝑍𝑋(𝐺𝑦)))
2120eqeq1d 2612 . . . . . . . 8 (((𝜑𝑦𝐴) ∧ 𝑥 = (𝐺𝑦)) → ((𝑍𝑋𝑥) = 𝑍 ↔ (𝑍𝑋(𝐺𝑦)) = 𝑍))
2218, 21rspcdv 3285 . . . . . . 7 ((𝜑𝑦𝐴) → (∀𝑥𝐵 (𝑍𝑋𝑥) = 𝑍 → (𝑍𝑋(𝐺𝑦)) = 𝑍))
2317, 22mpd 15 . . . . . 6 ((𝜑𝑦𝐴) → (𝑍𝑋(𝐺𝑦)) = 𝑍)
2423adantr 480 . . . . 5 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → (𝑍𝑋(𝐺𝑦)) = 𝑍)
2512, 14, 243eqtrd 2648 . . . 4 (((𝜑𝑦𝐴) ∧ (𝐹𝑦) = 𝑍) → ((𝐹𝑓 𝑋𝐺)‘𝑦) = 𝑍)
2625ex 449 . . 3 ((𝜑𝑦𝐴) → ((𝐹𝑦) = 𝑍 → ((𝐹𝑓 𝑋𝐺)‘𝑦) = 𝑍))
2726ralrimiva 2949 . 2 (𝜑 → ∀𝑦𝐴 ((𝐹𝑦) = 𝑍 → ((𝐹𝑓 𝑋𝐺)‘𝑦) = 𝑍))
283, 6, 7, 7, 8offn 6806 . . 3 (𝜑 → (𝐹𝑓 𝑋𝐺) Fn 𝐴)
29 ssid 3587 . . . 4 𝐴𝐴
3029a1i 11 . . 3 (𝜑𝐴𝐴)
31 suppofssd.2 . . 3 (𝜑𝑍𝐵)
32 suppfnss 7207 . . 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 1325 . 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 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793   supp csupp 7182 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-supp 7183 This theorem is referenced by:  frlmphllem  19938  rrxcph  22988
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