Theorem fvmptrabdm 48013

Description: Value of a function mapping a set to a class abstraction restricting the value of another function. See also fvmptrabfv 7024. (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		pm2.1 909	. 2 ⊢ (¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹)
3		imor 866	. . 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 6915	. . . . . . 7 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = ∅)
6		ndmfv 6915	. . . . . . . . 9 ⊢ (¬ 𝑌 ∈ dom 𝐺 → (𝐺‘𝑌) = ∅)
7	6	rabeqdv 3431	. . . . . . . 8 ⊢ (¬ 𝑌 ∈ dom 𝐺 → {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
8		rab0 4343	. . . . . . . 8 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅
9	7, 8	eqtr2di 2815	. . . . . . 7 ⊢ (¬ 𝑌 ∈ dom 𝐺 → ∅ = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
10	5, 9	sylan9eq 2818	. . . . . 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 3423	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
14	11	dmmpt 6243	. . . . . . . . . 10 ⊢ dom 𝐹 = {𝑥 ∈ 𝑉 ∣ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V}
15		rabid2 3449	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V} ↔ ∀𝑥 ∈ 𝑉 {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V)
16		fvex 6896	. . . . . . . . . . . . 13 ⊢ (𝐺‘𝑌) ∈ V
17	16	rabex 5311	. . . . . . . . . . . 12 ⊢ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V
18	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑉 → {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V)
19	15, 18	mprgbir 3086	. . . . . . . . . 10 ⊢ 𝑉 = {𝑥 ∈ 𝑉 ∣ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V}
20	14, 19	eqtr4i 2789	. . . . . . . . 9 ⊢ dom 𝐹 = 𝑉
21	20	eleq2i 2855	. . . . . . . 8 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝑉)
22	21	biimpi 219	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝑉)
23	16	rabex 5311	. . . . . . . 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 7015	. . . . . 6 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
26	10, 25	jaoi 870	. . . . 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 418	. . 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}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455  ∅c0 4287   ↦ cmpt 5193  dom cdm 5663  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fv 6546

This theorem is referenced by: (None)