Theorem fvmptrabfv 6838

Description: Value of a function mapping a set to a class abstraction restricting the value of another function. (Contributed by AV, 18-Feb-2022.)

Hypotheses

Ref	Expression
fvmptrabfv.f	⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑})
fvmptrabfv.r	⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
fvmptrabfv	⊢ (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}

Distinct variable groups:   𝑥,𝐺,𝑦   𝑥,𝑋,𝑦   𝜓,𝑥

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

Proof of Theorem fvmptrabfv

Step	Hyp	Ref	Expression
1		fveq2 6706	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
2		fvmptrabfv.r	. . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))
3	1, 2	rabeqbidv 3389	. . 3 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑} = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓})
4		fvmptrabfv.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑})
5		fvex 6719	. . . 4 ⊢ (𝐺‘𝑋) ∈ V
6	5	rabex 5214	. . 3 ⊢ {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} ∈ V
7	3, 4, 6	fvmpt 6807	. 2 ⊢ (𝑋 ∈ V → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓})
8		fvprc 6698	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
9		fvprc 6698	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐺‘𝑋) = ∅)
10	9	rabeqdv 3388	. . . 4 ⊢ (¬ 𝑋 ∈ V → {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
11		rab0 4287	. . . 4 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅
12	10, 11	eqtr2di 2791	. . 3 ⊢ (¬ 𝑋 ∈ V → ∅ = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓})
13	8, 12	eqtrd 2774	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓})
14	7, 13	pm2.61i 185	1 ⊢ (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1543   ∈ wcel 2110  {crab 3058  Vcvv 3401  ∅c0 4227   ↦ cmpt 5124  ‘cfv 6369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pr 5311

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-sbc 3688  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-nul 4228  df-if 4430  df-sn 4532  df-pr 4534  df-op 4538  df-uni 4810  df-br 5044  df-opab 5106  df-mpt 5125  df-id 5444  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-iota 6327  df-fun 6371  df-fv 6377

This theorem is referenced by:  uvtxval  27447