Theorem elmptrab 23963

Description: Membership in a one-parameter class of sets. (Contributed by Stefan O'Rear, 28-Jul-2015.)

Hypotheses

Ref	Expression
elmptrab.f	⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})
elmptrab.s1	⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓))
elmptrab.s2	⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
elmptrab.ex	⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
elmptrab	⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐶 ∧ 𝜓))

Distinct variable groups:   𝑥,𝑦,𝑋   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷   𝑥,𝑉,𝑦   𝑥,𝑌,𝑦   𝜓,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)   𝐷(𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem elmptrab

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmptrab.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑})
2	1	mptrcl 6999	. 2 ⊢ (𝑌 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐷)
3		simp1 1152	. 2 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐶 ∧ 𝜓) → 𝑋 ∈ 𝐷)
4		csbeq1 3855	. . . . . 6 ⊢ (𝑧 = 𝑋 → ⦋𝑧 / 𝑥⦌𝐵 = ⦋𝑋 / 𝑥⦌𝐵)
5		dfsbcq 3745	. . . . . 6 ⊢ (𝑧 = 𝑋 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑋 / 𝑥][𝑤 / 𝑦]𝜑))
6	4, 5	rabeqbidv 3432	. . . . 5 ⊢ (𝑧 = 𝑋 → {𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵 ∣ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑} = {𝑤 ∈ ⦋𝑋 / 𝑥⦌𝐵 ∣ [𝑋 / 𝑥][𝑤 / 𝑦]𝜑})
7		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑧{𝑦 ∈ 𝐵 ∣ 𝜑}
8		nfsbc1v 3763	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
9		nfcsb1v 3876	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
10	8, 9	nfrabw 3450	. . . . . . 7 ⊢ Ⅎ𝑥{𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵 ∣ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑}
11		csbeq1a 3866	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
12		sbceq1a 3754	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
13	11, 12	rabeqbidv 3432	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ ⦋𝑧 / 𝑥⦌𝐵 ∣ [𝑧 / 𝑥]𝜑})
14		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑤⦋𝑧 / 𝑥⦌𝐵
15		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑧 / 𝑥⦌𝐵
16		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑧
17		nfsbc1v 3763	. . . . . . . . . 10 ⊢ Ⅎ𝑦[𝑤 / 𝑦]𝜑
18	16, 17	nfsbcw 3765	. . . . . . . . 9 ⊢ Ⅎ𝑦[𝑧 / 𝑥][𝑤 / 𝑦]𝜑
19		nfv 1942	. . . . . . . . 9 ⊢ Ⅎ𝑤[𝑧 / 𝑥]𝜑
20		sbccom 3823	. . . . . . . . . 10 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
21		sbceq1a 3754	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑤 → ([𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
22	21	equcoms 2048	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
23	20, 22	bitr4id 293	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑥]𝜑))
24	14, 15, 18, 19, 23	cbvrabw 3449	. . . . . . . 8 ⊢ {𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵 ∣ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑} = {𝑦 ∈ ⦋𝑧 / 𝑥⦌𝐵 ∣ [𝑧 / 𝑥]𝜑}
25	13, 24	eqtr4di 2814	. . . . . . 7 ⊢ (𝑥 = 𝑧 → {𝑦 ∈ 𝐵 ∣ 𝜑} = {𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵 ∣ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑})
26	7, 10, 25	cbvmpt 5212	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 ↦ {𝑦 ∈ 𝐵 ∣ 𝜑}) = (𝑧 ∈ 𝐷 ↦ {𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵 ∣ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑})
27	1, 26	eqtri 2784	. . . . 5 ⊢ 𝐹 = (𝑧 ∈ 𝐷 ↦ {𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵 ∣ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑})
28		nfv 1942	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐷
29	9	nfel1 2939	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 ∈ 𝑉
30	28, 29	nfim 1924	. . . . . . 7 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐷 → ⦋𝑧 / 𝑥⦌𝐵 ∈ 𝑉)
31		eleq1w 2844	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐷 ↔ 𝑧 ∈ 𝐷))
32	11	eleq1d 2846	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐵 ∈ 𝑉 ↔ ⦋𝑧 / 𝑥⦌𝐵 ∈ 𝑉))
33	31, 32	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉) ↔ (𝑧 ∈ 𝐷 → ⦋𝑧 / 𝑥⦌𝐵 ∈ 𝑉)))
34		elmptrab.ex	. . . . . . 7 ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉)
35	30, 33, 34	chvarfv 2274	. . . . . 6 ⊢ (𝑧 ∈ 𝐷 → ⦋𝑧 / 𝑥⦌𝐵 ∈ 𝑉)
36		rabexg 5307	. . . . . 6 ⊢ (⦋𝑧 / 𝑥⦌𝐵 ∈ 𝑉 → {𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵 ∣ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑} ∈ V)
37	35, 36	syl 18	. . . . 5 ⊢ (𝑧 ∈ 𝐷 → {𝑤 ∈ ⦋𝑧 / 𝑥⦌𝐵 ∣ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑} ∈ V)
38	6, 27, 37	fvmpt3 6994	. . . 4 ⊢ (𝑋 ∈ 𝐷 → (𝐹‘𝑋) = {𝑤 ∈ ⦋𝑋 / 𝑥⦌𝐵 ∣ [𝑋 / 𝑥][𝑤 / 𝑦]𝜑})
39	38	eleq2d 2847	. . 3 ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ (𝐹‘𝑋) ↔ 𝑌 ∈ {𝑤 ∈ ⦋𝑋 / 𝑥⦌𝐵 ∣ [𝑋 / 𝑥][𝑤 / 𝑦]𝜑}))
40		dfsbcq 3745	. . . . . . 7 ⊢ (𝑤 = 𝑌 → ([𝑤 / 𝑦]𝜑 ↔ [𝑌 / 𝑦]𝜑))
41	40	sbcbidv 3798	. . . . . 6 ⊢ (𝑤 = 𝑌 → ([𝑋 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
42	41	elrab 3649	. . . . 5 ⊢ (𝑌 ∈ {𝑤 ∈ ⦋𝑋 / 𝑥⦌𝐵 ∣ [𝑋 / 𝑥][𝑤 / 𝑦]𝜑} ↔ (𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐵 ∧ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
43	42	a1i 11	. . . 4 ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ {𝑤 ∈ ⦋𝑋 / 𝑥⦌𝐵 ∣ [𝑋 / 𝑥][𝑤 / 𝑦]𝜑} ↔ (𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐵 ∧ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑)))
44		nfcvd 2924	. . . . . . 7 ⊢ (𝑋 ∈ 𝐷 → Ⅎ𝑥𝐶)
45		elmptrab.s2	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝐵 = 𝐶)
46	44, 45	csbiegf 3885	. . . . . 6 ⊢ (𝑋 ∈ 𝐷 → ⦋𝑋 / 𝑥⦌𝐵 = 𝐶)
47	46	eleq2d 2847	. . . . 5 ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐵 ↔ 𝑌 ∈ 𝐶))
48	47	anbi1d 642	. . . 4 ⊢ (𝑋 ∈ 𝐷 → ((𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐵 ∧ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑) ↔ (𝑌 ∈ 𝐶 ∧ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑)))
49		nfv 1942	. . . . . 6 ⊢ Ⅎ𝑥𝜓
50		nfv 1942	. . . . . 6 ⊢ Ⅎ𝑦𝜓
51		nfv 1942	. . . . . 6 ⊢ Ⅎ𝑥 𝑌 ∈ 𝐶
52		elmptrab.s1	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓))
53	49, 50, 51, 52	sbc2iegf 3817	. . . . 5 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐶) → ([𝑋 / 𝑥][𝑌 / 𝑦]𝜑 ↔ 𝜓))
54	53	pm5.32da 589	. . . 4 ⊢ (𝑋 ∈ 𝐷 → ((𝑌 ∈ 𝐶 ∧ [𝑋 / 𝑥][𝑌 / 𝑦]𝜑) ↔ (𝑌 ∈ 𝐶 ∧ 𝜓)))
55	43, 48, 54	3bitrd 308	. . 3 ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ {𝑤 ∈ ⦋𝑋 / 𝑥⦌𝐵 ∣ [𝑋 / 𝑥][𝑤 / 𝑦]𝜑} ↔ (𝑌 ∈ 𝐶 ∧ 𝜓)))
56		3anass 1109	. . . 4 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐶 ∧ 𝜓) ↔ (𝑋 ∈ 𝐷 ∧ (𝑌 ∈ 𝐶 ∧ 𝜓)))
57	56	baibr 545	. . 3 ⊢ (𝑋 ∈ 𝐷 → ((𝑌 ∈ 𝐶 ∧ 𝜓) ↔ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐶 ∧ 𝜓)))
58	39, 55, 57	3bitrd 308	. 2 ⊢ (𝑋 ∈ 𝐷 → (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐶 ∧ 𝜓)))
59	2, 3, 58	pm5.21nii 381	1 ⊢ (𝑌 ∈ (𝐹‘𝑋) ↔ (𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐶 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  {crab 3414  Vcvv 3453  [wsbc 3743  ⦋csb 3852   ↦ cmpt 5191  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  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 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fv 6544

This theorem is referenced by:  elmptrab2  23964  isfbas  23965