Theorem fsovfvfvd 44007

Description: Value of the operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵, when applied to function 𝐹 and element 𝑌. (Contributed by RP, 25-Apr-2021.)

Hypotheses

Ref	Expression
fsovd.fs	⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
fsovd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fsovd.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
fsovfvd.g	⊢ 𝐺 = (𝐴𝑂𝐵)
fsovfvd.f	⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑m 𝐴))
fsovfvfvd.h	⊢ 𝐻 = (𝐺‘𝐹)
fsovfvfvd.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
fsovfvfvd	⊢ (𝜑 → (𝐻‘𝑌) = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})

Distinct variable groups:   𝐴,𝑎,𝑏,𝑓,𝑥,𝑦   𝐵,𝑎,𝑏,𝑓,𝑦   𝑓,𝐹,𝑥,𝑦   𝑥,𝑌,𝑦   𝜑,𝑎,𝑏,𝑓,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑎,𝑏)   𝐺(𝑥,𝑦,𝑓,𝑎,𝑏)   𝐻(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑂(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑌(𝑓,𝑎,𝑏)

Proof of Theorem fsovfvfvd

Step	Hyp	Ref	Expression
1		fsovfvfvd.h	. . 3 ⊢ 𝐻 = (𝐺‘𝐹)
2		fsovd.fs	. . . 4 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))
3		fsovd.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		fsovd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
5		fsovfvd.g	. . . 4 ⊢ 𝐺 = (𝐴𝑂𝐵)
6		fsovfvd.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑m 𝐴))
7	2, 3, 4, 5, 6	fsovfvd 44006	. . 3 ⊢ (𝜑 → (𝐺‘𝐹) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))
8	1, 7	eqtrid 2777	. 2 ⊢ (𝜑 → 𝐻 = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))
9		eleq1 2817	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ (𝐹‘𝑥) ↔ 𝑌 ∈ (𝐹‘𝑥)))
10	9	rabbidv 3416	. . 3 ⊢ (𝑦 = 𝑌 → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})
11	10	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})
12		fsovfvfvd.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13		rabexg 5295	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)} ∈ V)
14	3, 13	syl 17	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)} ∈ V)
15	8, 11, 12, 14	fvmptd 6978	1 ⊢ (𝜑 → (𝐻‘𝑌) = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {crab 3408  Vcvv 3450  𝒫 cpw 4566   ↦ cmpt 5191  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392   ↑_m cmap 8802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395

This theorem is referenced by:  ntrneiel  44077