Theorem fvmptrabdm 47851

Description: Value of a function mapping a set to a class abstraction restricting the value of another function. See also fvmptrabfv 7004. (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		pm2.1 907	. 2 ⊢ (¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹)
3		imor 864	. . 3 ⊢ ((𝑌 ∈ dom 𝐺 → 𝑋 ∈ dom 𝐹) ↔ (¬ 𝑌 ∈ dom 𝐺 ∨ 𝑋 ∈ dom 𝐹))
4		ordir 1019	. . . . 5 ⊢ (((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) ∨ 𝑋 ∈ dom 𝐹) ↔ ((¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹) ∧ (¬ 𝑌 ∈ dom 𝐺 ∨ 𝑋 ∈ dom 𝐹)))
5		ndmfv 6895	. . . . . . 7 ⊢ (¬ 𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = ∅)
6		ndmfv 6895	. . . . . . . . 9 ⊢ (¬ 𝑌 ∈ dom 𝐺 → (𝐺‘𝑌) = ∅)
7	6	rabeqdv 3428	. . . . . . . 8 ⊢ (¬ 𝑌 ∈ dom 𝐺 → {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓})
8		rab0 4338	. . . . . . . 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 6223	. . . . . . . . . 10 ⊢ dom 𝐹 = {𝑥 ∈ 𝑉 ∣ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V}
15		rabid2 3446	. . . . . . . . . . 11 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V} ↔ ∀𝑥 ∈ 𝑉 {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V)
16		fvex 6876	. . . . . . . . . . . . 13 ⊢ (𝐺‘𝑌) ∈ V
17	16	rabex 5294	. . . . . . . . . . . 12 ⊢ {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V
18	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑉 → {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜑} ∈ V)
19	15, 18	mprgbir 3082	. . . . . . . . . 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 218	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ 𝑉)
23	16	rabex 5294	. . . . . . . 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 6995	. . . . . 6 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
26	10, 25	jaoi 868	. . . . 5 ⊢ (((¬ 𝑋 ∈ dom 𝐹 ∧ ¬ 𝑌 ∈ dom 𝐺) ∨ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
27	4, 26	sylbir 237	. . . 4 ⊢ (((¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹) ∧ (¬ 𝑌 ∈ dom 𝐺 ∨ 𝑋 ∈ dom 𝐹)) → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓})
28	27	expcom 417	. . 3 ⊢ ((¬ 𝑌 ∈ dom 𝐺 ∨ 𝑋 ∈ dom 𝐹) → ((¬ 𝑋 ∈ dom 𝐹 ∨ 𝑋 ∈ dom 𝐹) → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑌) ∣ 𝜓}))
29	3, 28	sylbi 219	. 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 208   ∧ wa 399   ∨ wo 858   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453  ∅c0 4285   ↦ cmpt 5180  dom cdm 5645  ‘cfv 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fv 6525

This theorem is referenced by: (None)