Theorem fvmptrab 43484

Description: Value of a function mapping a set to a class abstraction restricting a class depending on the argument of the function. More general version of fvmptrabfv 6794, but relying on the fact that out-of-domain arguments evaluate to the empty set, which relies on set.mm's particular encoding. (Contributed by AV, 14-Feb-2022.)

Hypotheses

Ref	Expression
fvmptrab.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑})
fvmptrab.r	⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))
fvmptrab.s	⊢ (𝑥 = 𝑋 → 𝑀 = 𝑁)
fvmptrab.v	⊢ (𝑋 ∈ 𝑉 → 𝑁 ∈ V)
fvmptrab.n	⊢ (𝑋 ∉ 𝑉 → 𝑁 = ∅)

Assertion

Ref	Expression
fvmptrab	⊢ (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓}

Distinct variable groups:   𝑦,𝑀   𝑥,𝑁,𝑦   𝑥,𝑋,𝑦   𝑥,𝑉   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝐹(𝑥,𝑦)   𝑀(𝑥)   𝑉(𝑦)

Proof of Theorem fvmptrab

Step	Hyp	Ref	Expression
1		fvmptrab.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑})
2	1	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑}))
3		fvmptrab.s	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝑀 = 𝑁)
4		fvmptrab.r	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))
5	3, 4	rabeqbidv 3486	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ 𝑁 ∣ 𝜓})
6	5	adantl 484	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ 𝑁 ∣ 𝜓})
7		id 22	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝑉)
8		eqid 2821	. . . 4 ⊢ {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ 𝑁 ∣ 𝜓}
9		fvmptrab.v	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑁 ∈ V)
10	8, 9	rabexd 5229	. . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑦 ∈ 𝑁 ∣ 𝜓} ∈ V)
11	2, 6, 7, 10	fvmptd 6770	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓})
12	1	fvmptndm 6793	. . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝐹‘𝑋) = ∅)
13		df-nel 3124	. . . 4 ⊢ (𝑋 ∉ 𝑉 ↔ ¬ 𝑋 ∈ 𝑉)
14		fvmptrab.n	. . . . 5 ⊢ (𝑋 ∉ 𝑉 → 𝑁 = ∅)
15		rabeq 3484	. . . . . 6 ⊢ (𝑁 = ∅ → {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
16		rab0 4337	. . . . . 6 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅
17	15, 16	syl6req 2873	. . . . 5 ⊢ (𝑁 = ∅ → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓})
18	14, 17	syl 17	. . . 4 ⊢ (𝑋 ∉ 𝑉 → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓})
19	13, 18	sylbir 237	. . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓})
20	12, 19	eqtrd 2856	. 2 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓})
21	11, 20	pm2.61i 184	1 ⊢ (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110   ∉ wnel 3123  {crab 3142  Vcvv 3495  ∅c0 4291   ↦ cmpt 5139  ‘cfv 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  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-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-iota 6309  df-fun 6352  df-fv 6358

This theorem is referenced by: (None)