Theorem suppvalfng 7849

Description: The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 7850 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5161. (Contributed by SN, 5-Aug-2024.)

Assertion

Ref	Expression
suppvalfng	⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})

Distinct variable groups:   𝑖,𝑉   𝑖,𝑊   𝑖,𝑋   𝑖,𝑍   𝑖,𝐹

Proof of Theorem suppvalfng

Step	Hyp	Ref	Expression
1		fnfun 6440	. . 3 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹)
2		suppval1 7848	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍})
3	1, 2	syl3an1 1161	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍})
4		fndm 6442	. . . 4 ⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋)
5	4	3ad2ant1 1131	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → dom 𝐹 = 𝑋)
6	5	rabeqdv 3398	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑖 ∈ dom 𝐹 ∣ (𝐹‘𝑖) ≠ 𝑍} = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})
7	3, 6	eqtrd 2794	1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})

Syntax hints:   → wi 4   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  {crab 3075  dom cdm 5529  Fun wfun 6335   Fn wfn 6336  ‘cfv 6341  (class class class)co 7157   supp csupp 7842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303  ax-un 7466

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3700  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-br 5038  df-opab 5100  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6300  df-fun 6343  df-fn 6344  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-supp 7843

This theorem is referenced by:  elsuppfng  7851