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Theorem suppcofnd 7846
Description: The support of the composition of two functions. (Contributed by SN, 15-Sep-2023.)
Hypotheses
Ref Expression
suppcofnd.z (𝜑𝑍𝑈)
suppcofnd.f (𝜑𝐹 Fn 𝐴)
suppcofnd.a (𝜑𝐴𝑉)
suppcofnd.g (𝜑𝐺 Fn 𝐵)
suppcofnd.b (𝜑𝐵𝑊)
Assertion
Ref Expression
suppcofnd (𝜑 → ((𝐹𝐺) supp 𝑍) = {𝑥𝐵 ∣ ((𝐺𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺𝑥)) ≠ 𝑍)})
Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝑍   𝑥,𝑈   𝑥,𝑊   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem suppcofnd
StepHypRef Expression
1 suppcofnd.f . . . 4 (𝜑𝐹 Fn 𝐴)
2 suppcofnd.a . . . 4 (𝜑𝐴𝑉)
31, 2fnexd 6954 . . 3 (𝜑𝐹 ∈ V)
4 suppcofnd.g . . . 4 (𝜑𝐺 Fn 𝐵)
5 suppcofnd.b . . . 4 (𝜑𝐵𝑊)
64, 5fnexd 6954 . . 3 (𝜑𝐺 ∈ V)
7 suppco 7845 . . 3 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
83, 6, 7syl2anc 587 . 2 (𝜑 → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
9 fncnvima2 6804 . . 3 (𝐺 Fn 𝐵 → (𝐺 “ (𝐹 supp 𝑍)) = {𝑥𝐵 ∣ (𝐺𝑥) ∈ (𝐹 supp 𝑍)})
104, 9syl 17 . 2 (𝜑 → (𝐺 “ (𝐹 supp 𝑍)) = {𝑥𝐵 ∣ (𝐺𝑥) ∈ (𝐹 supp 𝑍)})
11 suppcofnd.z . . . 4 (𝜑𝑍𝑈)
12 elsuppfn 7813 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉𝑍𝑈) → ((𝐺𝑥) ∈ (𝐹 supp 𝑍) ↔ ((𝐺𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺𝑥)) ≠ 𝑍)))
131, 2, 11, 12syl3anc 1368 . . 3 (𝜑 → ((𝐺𝑥) ∈ (𝐹 supp 𝑍) ↔ ((𝐺𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺𝑥)) ≠ 𝑍)))
1413rabbidv 3457 . 2 (𝜑 → {𝑥𝐵 ∣ (𝐺𝑥) ∈ (𝐹 supp 𝑍)} = {𝑥𝐵 ∣ ((𝐺𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺𝑥)) ≠ 𝑍)})
158, 10, 143eqtrd 2860 1 (𝜑 → ((𝐹𝐺) supp 𝑍) = {𝑥𝐵 ∣ ((𝐺𝑥) ∈ 𝐴 ∧ (𝐹‘(𝐺𝑥)) ≠ 𝑍)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3007  {crab 3130  Vcvv 3471  ccnv 5527  cima 5531  ccom 5532   Fn wfn 6323  cfv 6328  (class class class)co 7130   supp csupp 7805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-supp 7806
This theorem is referenced by:  mhpinvcl  20315
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