Theorem fvmptrab 47884

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 7012, 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 3435	. . . 4 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ 𝑁 ∣ 𝜓})
6	5	adantl 486	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑥 = 𝑋) → {𝑦 ∈ 𝑀 ∣ 𝜑} = {𝑦 ∈ 𝑁 ∣ 𝜓})
7		id 23	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝑉)
8		eqid 2765	. . . 4 ⊢ {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ 𝑁 ∣ 𝜓}
9		fvmptrab.v	. . . 4 ⊢ (𝑋 ∈ 𝑉 → 𝑁 ∈ V)
10	8, 9	rabexd 5301	. . 3 ⊢ (𝑋 ∈ 𝑉 → {𝑦 ∈ 𝑁 ∣ 𝜓} ∈ V)
11	2, 6, 7, 10	fvmptd 6987	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝐹‘𝑋) = {𝑦 ∈ 𝑁 ∣ 𝜓})
12	1	fvmptndm 7011	. . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → (𝐹‘𝑋) = ∅)
13		df-nel 3065	. . . 4 ⊢ (𝑋 ∉ 𝑉 ↔ ¬ 𝑋 ∈ 𝑉)
14		fvmptrab.n	. . . . 5 ⊢ (𝑋 ∉ 𝑉 → 𝑁 = ∅)
15		rabeq 3431	. . . . . 6 ⊢ (𝑁 = ∅ → {𝑦 ∈ 𝑁 ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
16		rab0 4342	. . . . . 6 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅
17	15, 16	eqtr2di 2817	. . . . 5 ⊢ (𝑁 = ∅ → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓})
18	14, 17	syl 18	. . . 4 ⊢ (𝑋 ∉ 𝑉 → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓})
19	13, 18	sylbir 238	. . 3 ⊢ (¬ 𝑋 ∈ 𝑉 → ∅ = {𝑦 ∈ 𝑁 ∣ 𝜓})
20	12, 19	eqtrd 2800	. 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 209   = wceq 1563   ∈ wcel 2145   ∉ wnel 3064  {crab 3417  Vcvv 3457  ∅c0 4288   ↦ cmpt 5186  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533

This theorem is referenced by: (None)