Theorem fvmptrabdm 47535

Description: Value of a function mapping a set to a class abstraction restricting the value of another function. See also fvmptrabfv 6973. (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		pm2.1 896	. 2 ⊢ (¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹)
3		imor 853	. . 3 ⊢ ((𝑌 ∈ dom 𝐺 → 𝑋 ∈ dom 𝐹) ↔ (¬ 𝑌 ∈ dom 𝐺 ∨ 𝑋 ∈ dom 𝐹))
4		ordir 1008	. . . . 5 ⊢ (((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) ∨ 𝑋 ∈ dom 𝐹) ↔ ((¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹) ∧ (¬ 𝑌 ∈ dom 𝐺 ∨ 𝑋 ∈ dom 𝐹)))
5		ndmfv 6866	. . . . . . 7 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = ∅)
6		ndmfv 6866	. . . . . . . . 9 ⊢ (¬ 𝑌 ∈ dom 𝐺 → (𝐺‘𝑌) = ∅)
7	6	rabeqdv 3414	. . . . . . . 8 ⊢ (¬ 𝑌 ∈ dom 𝐺 → {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
8		rab0 4338	. . . . . . . 8 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅
9	7, 8	eqtr2di 2788	. . . . . . 7 ⊢ (¬ 𝑌 ∈ dom 𝐺 → ∅ = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
10	5, 9	sylan9eq 2791	. . . . . 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 3406	. . . . . . 7 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
14	11	dmmpt 6198	. . . . . . . . . 10 ⊢ dom 𝐹 = {𝑥 ∈ 𝑉 ∣ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V}
15		rabid2 3432	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V} ↔ ∀𝑥 ∈ 𝑉 {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V)
16		fvex 6847	. . . . . . . . . . . . 13 ⊢ (𝐺‘𝑌) ∈ V
17	16	rabex 5284	. . . . . . . . . . . 12 ⊢ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V
18	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑉 → {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V)
19	15, 18	mprgbir 3058	. . . . . . . . . 10 ⊢ 𝑉 = {𝑥 ∈ 𝑉 ∣ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V}
20	14, 19	eqtr4i 2762	. . . . . . . . 9 ⊢ dom 𝐹 = 𝑉
21	20	eleq2i 2828	. . . . . . . 8 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝑉)
22	21	biimpi 216	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝑉)
23	16	rabex 5284	. . . . . . . 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 6964	. . . . . 6 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
26	10, 25	jaoi 857	. . . . 5 ⊢ (((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) ∨ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
27	4, 26	sylbir 235	. . . 4 ⊢ (((¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹) ∧ (¬ 𝑌 ∈ dom 𝐺 ∨ 𝑋 ∈ dom 𝐹)) → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
28	27	expcom 413	. . 3 ⊢ ((¬ 𝑌 ∈ dom 𝐺 ∨ 𝑋 ∈ dom 𝐹) → ((¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓}))
29	3, 28	sylbi 217	. 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 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440  ∅c0 4285   ↦ cmpt 5179  dom cdm 5624  ‘cfv 6492

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500

This theorem is referenced by: (None)