Theorem fvmptrab 47207

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 7061, 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 3462	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ 𝑁 ∣ 𝜓})
6	5	adantl 481	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ 𝑁 ∣ 𝜓})
7		id 22	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝑉)
8		eqid 2740	. . . 4 ⊢ {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ 𝑁 ∣ 𝜓}
9		fvmptrab.v	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑁 ∈ V)
10	8, 9	rabexd 5358	. . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑦 ∈ 𝑁 ∣ 𝜓} ∈ V)
11	2, 6, 7, 10	fvmptd 7036	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓})
12	1	fvmptndm 7060	. . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝐹‘𝑋) = ∅)
13		df-nel 3053	. . . 4 ⊢ (𝑋 ∉ 𝑉 ↔ ¬ 𝑋 ∈ 𝑉)
14		fvmptrab.n	. . . . 5 ⊢ (𝑋 ∉ 𝑉 → 𝑁 = ∅)
15		rabeq 3458	. . . . . 6 ⊢ (𝑁 = ∅ → {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
16		rab0 4409	. . . . . 6 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅
17	15, 16	eqtr2di 2797	. . . . 5 ⊢ (𝑁 = ∅ → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓})
18	14, 17	syl 17	. . . 4 ⊢ (𝑋 ∉ 𝑉 → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓})
19	13, 18	sylbir 235	. . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓})
20	12, 19	eqtrd 2780	. 2 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓})
21	11, 20	pm2.61i 182	1 ⊢ (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108   ∉ wnel 3052  {crab 3443  Vcvv 3488  ∅c0 4352   ↦ cmpt 5249  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581

This theorem is referenced by: (None)