Theorem mpoxopoveq 6137

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

Assertion

Ref	Expression
mpoxopoveq	⊢ (((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 mpoxopoveq

Step	Hyp	Ref	Expression
1		mpoxopoveq.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 5421	. . . . 5 ⊢ (𝑥 = ⟨𝑉, 𝑊⟩ → (1st ‘𝑥) = (1st ‘⟨𝑉, 𝑊⟩))
4		op1stg 6048	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
5	4	adantr 274	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
6	3, 5	sylan9eqr 2194	. . . 4 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ 𝑥 = ⟨𝑉, 𝑊⟩) → (1st ‘𝑥) = 𝑉)
7	6	adantrr 470	. . 3 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (1st ‘𝑥) = 𝑉)
8		sbceq1a 2918	. . . . . 6 ⊢ (𝑦 = 𝐾 → (𝜑 ↔ [𝐾 / 𝑦]𝜑))
9	8	adantl 275	. . . . 5 ⊢ ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → (𝜑 ↔ [𝐾 / 𝑦]𝜑))
10	9	adantl 275	. . . 4 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑 ↔ [𝐾 / 𝑦]𝜑))
11		sbceq1a 2918	. . . . . 6 ⊢ (𝑥 = ⟨𝑉, 𝑊⟩ → ([𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
12	11	adantr 274	. . . . 5 ⊢ ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → ([𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
13	12	adantl 275	. . . 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 187	. . 3 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
15	7, 14	rabeqbidv 2681	. 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 4150	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → ⟨𝑉, 𝑊⟩ ∈ V)
17	16	adantr 274	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → ⟨𝑉, 𝑊⟩ ∈ V)
18		simpr 109	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → 𝐾 ∈ 𝑉)
19		rabexg 4071	. . 3 ⊢ (𝑉 ∈ 𝑋 → {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
20	19	ad2antrr 479	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
21		equid 1677	. . 3 ⊢ 𝑧 = 𝑧
22		nfvd 1509	. . 3 ⊢ (𝑧 = 𝑧 → Ⅎ𝑥((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉))
23	21, 22	ax-mp 5	. 2 ⊢ Ⅎ𝑥((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉)
24		nfvd 1509	. . 3 ⊢ (𝑧 = 𝑧 → Ⅎ𝑦((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉))
25	21, 24	ax-mp 5	. 2 ⊢ Ⅎ𝑦((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉)
26		nfcv 2281	. 2 ⊢ Ⅎ𝑦⟨𝑉, 𝑊⟩
27		nfcv 2281	. 2 ⊢ Ⅎ𝑥𝐾
28		nfsbc1v 2927	. . 3 ⊢ Ⅎ𝑥[⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
29		nfcv 2281	. . 3 ⊢ Ⅎ𝑥𝑉
30	28, 29	nfrabxy 2611	. 2 ⊢ Ⅎ𝑥{𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
31		nfsbc1v 2927	. . . 4 ⊢ Ⅎ𝑦[𝐾 / 𝑦]𝜑
32	26, 31	nfsbc 2929	. . 3 ⊢ Ⅎ𝑦[⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
33		nfcv 2281	. . 3 ⊢ Ⅎ𝑦𝑉
34	32, 33	nfrabxy 2611	. 2 ⊢ Ⅎ𝑦{𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
35	2, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34	ovmpodxf 5896	1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  Ⅎwnf 1436   ∈ wcel 1480  {crab 2420  Vcvv 2686  [wsbc 2909  ⟨cop 3530  ‘cfv 5123  (class class class)co 5774   ∈ cmpo 5776  1^st c1st 6036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038

This theorem is referenced by:  mpoxopovel  6138