Theorem fsovfvfvd 40712

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 40711	. . 3 ⊢ (𝜑 → (𝐺‘𝐹) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))
8	1, 7	syl5eq 2845	. 2 ⊢ (𝜑 → 𝐻 = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))
9		eleq1 2877	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ (𝐹‘𝑥) ↔ 𝑌 ∈ (𝐹‘𝑥)))
10	9	rabbidv 3427	. . 3 ⊢ (𝑦 = 𝑌 → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})
11	10	adantl 485	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})
12		fsovfvfvd.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13		rabexg 5198	. . 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 6752	1 ⊢ (𝜑 → (𝐻‘𝑌) = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {crab 3110  Vcvv 3441  𝒫 cpw 4497   ↦ cmpt 5110  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   ↑_m cmap 8389

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140

This theorem is referenced by:  ntrneiel  40784