Theorem fvmptrabfv 7031

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 6891	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
2		fvmptrabfv.r	. . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))
3	1, 2	rabeqbidv 3437	. . 3 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑} = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓})
4		fvmptrabfv.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑})
5		fvex 6904	. . . 4 ⊢ (𝐺‘𝑋) ∈ V
6	5	rabex 5329	. . 3 ⊢ {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} ∈ V
7	3, 4, 6	fvmpt 6999	. 2 ⊢ (𝑋 ∈ V → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓})
8		fvprc 6883	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
9		fvprc 6883	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐺‘𝑋) = ∅)
10	9	rabeqdv 3435	. . . 4 ⊢ (¬ 𝑋 ∈ V → {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
11		rab0 4378	. . . 4 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅
12	10, 11	eqtr2di 2782	. . 3 ⊢ (¬ 𝑋 ∈ V → ∅ = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓})
13	8, 12	eqtrd 2765	. 2 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓})
14	7, 13	pm2.61i 182	1 ⊢ (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {crab 3419  Vcvv 3463  ∅c0 4318   ↦ cmpt 5226  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550

This theorem is referenced by:  uvtxval  29242