Theorem fvmptrabdm 48307

Description: Value of a function mapping a set to a class abstraction restricting the value of another function. See also fvmptrabfv 7018. (Suggested by BJ, 18-Feb-2022.) (Contributed by AV, 18-Feb-2022.)

Hypotheses

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

Assertion

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

Distinct variable groups:   𝑥,𝐹   𝑥,𝐺,𝑦   𝑥,𝑉   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜓,𝑥

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

Proof of Theorem fvmptrabdm

Step	Hyp	Ref	Expression
1		fvmptrabdm.v	. 2 ⊢ (𝑌 ∈ dom 𝐺 → 𝑋 ∈ dom 𝐹)
2		pm2.1 910	. 2 ⊢ (¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹)
3		imor 867	. . 3 ⊢ ((𝑌 ∈ dom 𝐺 → 𝑋 ∈ dom 𝐹) ↔ (¬ 𝑌 ∈ dom 𝐺 ∨ 𝑋 ∈ dom 𝐹))
4		ordir 1024	. . . . 5 ⊢ (((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) ∨ 𝑋 ∈ dom 𝐹) ↔ ((¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹) ∧ (¬ 𝑌 ∈ dom 𝐺 ∨ 𝑋 ∈ dom 𝐹)))
5		ndmfv 6909	. . . . . . 7 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = ∅)
6		ndmfv 6909	. . . . . . . . 9 ⊢ (¬ 𝑌 ∈ dom 𝐺 → (𝐺‘𝑌) = ∅)
7	6	rabeqdv 3428	. . . . . . . 8 ⊢ (¬ 𝑌 ∈ dom 𝐺 → {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
8		rab0 4335	. . . . . . . 8 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅
9	7, 8	eqtr2di 2813	. . . . . . 7 ⊢ (¬ 𝑌 ∈ dom 𝐺 → ∅ = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
10	5, 9	sylan9eq 2816	. . . . . 6 ⊢ ((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
11		fvmptrabdm.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑})
12		fvmptrabdm.r	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))
13	12	rabbidv 3420	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
14	11	dmmpt 6234	. . . . . . . . . 10 ⊢ dom 𝐹 = {𝑥 ∈ 𝑉 ∣ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V}
15		rabid2 3445	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V} ↔ ∀𝑥 ∈ 𝑉 {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V)
16		fvex 6890	. . . . . . . . . . . . 13 ⊢ (𝐺‘𝑌) ∈ V
17	16	rabex 5300	. . . . . . . . . . . 12 ⊢ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V
18	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑉 → {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V)
19	15, 18	mprgbir 3084	. . . . . . . . . 10 ⊢ 𝑉 = {𝑥 ∈ 𝑉 ∣ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V}
20	14, 19	eqtr4i 2787	. . . . . . . . 9 ⊢ dom 𝐹 = 𝑉
21	20	eleq2i 2853	. . . . . . . 8 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝑉)
22	21	biimpi 219	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝑉)
23	16	rabex 5300	. . . . . . . 8 ⊢ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓} ∈ V
24	23	a1i 11	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐹 → {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓} ∈ V)
25	11, 13, 22, 24	fvmptd3 7009	. . . . . 6 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
26	10, 25	jaoi 871	. . . . 5 ⊢ (((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) ∨ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
27	4, 26	sylbir 238	. . . 4 ⊢ (((¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹) ∧ (¬ 𝑌 ∈ dom 𝐺 ∨ 𝑋 ∈ dom 𝐹)) → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
28	27	expcom 419	. . 3 ⊢ ((¬ 𝑌 ∈ dom 𝐺 ∨ 𝑋 ∈ dom 𝐹) → ((¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓}))
29	3, 28	sylbi 220	. 2 ⊢ ((𝑌 ∈ dom 𝐺 → 𝑋 ∈ dom 𝐹) → ((¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓}))
30	1, 2, 29	mp2 9	1 ⊢ (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓}

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  {crab 3413  Vcvv 3451  ∅c0 4279   ↦ cmpt 5186  dom cdm 5651  ‘cfv 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fv 6539

This theorem is used by: (None)