Theorem fvmptrabfv 6967

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 6828	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
2		fvmptrabfv.r	. . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))
3	1, 2	rabeqbidv 3414	. . 3 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑} = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓})
4		fvmptrabfv.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑})
5		fvex 6841	. . . 4 ⊢ (𝐺‘𝑋) ∈ V
6	5	rabex 5279	. . 3 ⊢ {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} ∈ V
7	3, 4, 6	fvmpt 6935	. 2 ⊢ (𝑋 ∈ V → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓})
8		fvprc 6820	. . 3 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
9		fvprc 6820	. . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐺‘𝑋) = ∅)
10	9	rabeqdv 3411	. . . 4 ⊢ (¬ 𝑋 ∈ V → {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
11		rab0 4335	. . . 4 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅
12	10, 11	eqtr2di 2785	. . 3 ⊢ (¬ 𝑋 ∈ V → ∅ = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓})
13	8, 12	eqtrd 2768	. 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 206   = wceq 1541   ∈ wcel 2113  {crab 3396  Vcvv 3437  ∅c0 4282   ↦ cmpt 5174  ‘cfv 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494

This theorem is referenced by:  uvtxval  29367