Theorem fsovfvfvd 40377

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 40376	. . 3 ⊢ (𝜑 → (𝐺‘𝐹) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))
8	1, 7	syl5eq 2868	. 2 ⊢ (𝜑 → 𝐻 = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)}))
9		eleq1 2900	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ∈ (𝐹‘𝑥) ↔ 𝑌 ∈ (𝐹‘𝑥)))
10	9	rabbidv 3480	. . 3 ⊢ (𝑦 = 𝑌 → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})
11	10	adantl 484	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝑌) → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝐹‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})
12		fsovfvfvd.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13		rabexg 5234	. . 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 6775	1 ⊢ (𝜑 → (𝐻‘𝑌) = {𝑥 ∈ 𝐴 ∣ 𝑌 ∈ (𝐹‘𝑥)})

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  {crab 3142  Vcvv 3494  𝒫 cpw 4539   ↦ cmpt 5146  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158   ↑_m cmap 8406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161

This theorem is referenced by:  ntrneiel  40451