Theorem fsovfvfvd 43338

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 43337	. . 3 ⊢ (𝜑 → (𝐺‘𝐹) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))
8	1, 7	eqtrid 2778	. 2 ⊢ (𝜑 → 𝐻 = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))
9		eleq1 2815	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ (𝐹‘𝑥) ↔ 𝑌 ∈ (𝐹‘𝑥)))
10	9	rabbidv 3434	. . 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 5324	. . 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 6999	1 ⊢ (𝜑 → (𝐻‘𝑌) = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468  𝒫 cpw 4597   ↦ cmpt 5224  ‘cfv 6537  (class class class)co 7405   ∈ cmpo 7407   ↑_m cmap 8822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410

This theorem is referenced by:  ntrneiel  43408