Theorem mpt2xopoveq 5878

Description: Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)

Hypothesis

Ref	Expression
mpt2xopoveq.f	⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})

Assertion

Ref	Expression
mpt2xopoveq	⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})

Distinct variable groups:   𝑛,𝐾,𝑥,𝑦   𝑛,𝑉,𝑥,𝑦   𝑛,𝑊,𝑥,𝑦   𝑛,𝑋,𝑥,𝑦   𝑛,𝑌,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑛)   𝐹(𝑥,𝑦,𝑛)

Proof of Theorem mpt2xopoveq

Step	Hyp	Ref	Expression
1		mpt2xopoveq.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})
2	1	a1i 9	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}))
3		fveq2 5198	. . . . 5 ⊢ (𝑥 = ⟨𝑉, 𝑊⟩ → (1st ‘𝑥) = (1st ‘⟨𝑉, 𝑊⟩))
4		op1stg 5797	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
5	4	adantr 270	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
6	3, 5	sylan9eqr 2135	. . . 4 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ 𝑥 = ⟨𝑉, 𝑊⟩) → (1st ‘𝑥) = 𝑉)
7	6	adantrr 462	. . 3 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (1st ‘𝑥) = 𝑉)
8		sbceq1a 2824	. . . . . 6 ⊢ (𝑦 = 𝐾 → (𝜑 ↔ [𝐾 / 𝑦]𝜑))
9	8	adantl 271	. . . . 5 ⊢ ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → (𝜑 ↔ [𝐾 / 𝑦]𝜑))
10	9	adantl 271	. . . 4 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑 ↔ [𝐾 / 𝑦]𝜑))
11		sbceq1a 2824	. . . . . 6 ⊢ (𝑥 = ⟨𝑉, 𝑊⟩ → ([𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
12	11	adantr 270	. . . . 5 ⊢ ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → ([𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
13	12	adantl 271	. . . 4 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → ([𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
14	10, 13	bitrd 186	. . 3 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
15	7, 14	rabeqbidv 2596	. 2 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑} = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
16		opexg 3983	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → ⟨𝑉, 𝑊⟩ ∈ V)
17	16	adantr 270	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → ⟨𝑉, 𝑊⟩ ∈ V)
18		simpr 108	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → 𝐾 ∈ 𝑉)
19		rabexg 3921	. . 3 ⊢ (𝑉 ∈ 𝑋 → {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
20	19	ad2antrr 471	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
21		equid 1629	. . 3 ⊢ 𝑧 = 𝑧
22		nfvd 1462	. . 3 ⊢ (𝑧 = 𝑧 → Ⅎ𝑥((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉))
23	21, 22	ax-mp 7	. 2 ⊢ Ⅎ𝑥((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉)
24		nfvd 1462	. . 3 ⊢ (𝑧 = 𝑧 → Ⅎ𝑦((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉))
25	21, 24	ax-mp 7	. 2 ⊢ Ⅎ𝑦((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉)
26		nfcv 2219	. 2 ⊢ Ⅎ𝑦⟨𝑉, 𝑊⟩
27		nfcv 2219	. 2 ⊢ Ⅎ𝑥𝐾
28		nfsbc1v 2833	. . 3 ⊢ Ⅎ𝑥[⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
29		nfcv 2219	. . 3 ⊢ Ⅎ𝑥𝑉
30	28, 29	nfrabxy 2534	. 2 ⊢ Ⅎ𝑥{𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
31		nfsbc1v 2833	. . . 4 ⊢ Ⅎ𝑦[𝐾 / 𝑦]𝜑
32	26, 31	nfsbc 2835	. . 3 ⊢ Ⅎ𝑦[⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
33		nfcv 2219	. . 3 ⊢ Ⅎ𝑦𝑉
34	32, 33	nfrabxy 2534	. 2 ⊢ Ⅎ𝑦{𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
35	2, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34	ovmpt2dxf 5646	1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  Ⅎwnf 1389   ∈ wcel 1433  {crab 2352  Vcvv 2601  [wsbc 2815  ⟨cop 3401  ‘cfv 4922  (class class class)co 5532   ↦ cmpt2 5534  1^st c1st 5785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787

This theorem is referenced by:  mpt2xopovel  5879