Theorem mpoxopoveq 8260

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 11	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}))
3		fveq2 6920	. . . . 5 ⊢ (𝑥 = ⟨𝑉, 𝑊⟩ → (1st ‘𝑥) = (1st ‘⟨𝑉, 𝑊⟩))
4		op1stg 8042	. . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
5	4	adantr 480	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (1st ‘⟨𝑉, 𝑊⟩) = 𝑉)
6	3, 5	sylan9eqr 2802	. . . 4 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ 𝑥 = ⟨𝑉, 𝑊⟩) → (1st ‘𝑥) = 𝑉)
7	6	adantrr 716	. . 3 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (1st ‘𝑥) = 𝑉)
8		sbceq1a 3815	. . . . . 6 ⊢ (𝑦 = 𝐾 → (𝜑 ↔ [𝐾 / 𝑦]𝜑))
9	8	adantl 481	. . . . 5 ⊢ ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → (𝜑 ↔ [𝐾 / 𝑦]𝜑))
10	9	adantl 481	. . . 4 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑 ↔ [𝐾 / 𝑦]𝜑))
11		sbceq1a 3815	. . . . . 6 ⊢ (𝑥 = ⟨𝑉, 𝑊⟩ → ([𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
12	11	adantr 480	. . . . 5 ⊢ ((𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾) → ([𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
13	12	adantl 481	. . . 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 279	. . 3 ⊢ ((((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) ∧ (𝑥 = ⟨𝑉, 𝑊⟩ ∧ 𝑦 = 𝐾)) → (𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
15	7, 14	rabeqbidv 3462	. 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		opex 5484	. . 3 ⊢ ⟨𝑉, 𝑊⟩ ∈ V
17	16	a1i 11	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → ⟨𝑉, 𝑊⟩ ∈ V)
18		simpr 484	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → 𝐾 ∈ 𝑉)
19		rabexg 5355	. . 3 ⊢ (𝑉 ∈ 𝑋 → {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
20	19	ad2antrr 725	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ∈ V)
21		equid 2011	. . 3 ⊢ 𝑧 = 𝑧
22		nfvd 1914	. . 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 1914	. . 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 2908	. 2 ⊢ Ⅎ𝑦⟨𝑉, 𝑊⟩
27		nfcv 2908	. 2 ⊢ Ⅎ𝑥𝐾
28		nfsbc1v 3824	. . 3 ⊢ Ⅎ𝑥[⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
29		nfcv 2908	. . 3 ⊢ Ⅎ𝑥𝑉
30	28, 29	nfrabw 3483	. 2 ⊢ Ⅎ𝑥{𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
31		nfsbc1v 3824	. . . 4 ⊢ Ⅎ𝑦[𝐾 / 𝑦]𝜑
32	26, 31	nfsbcw 3826	. . 3 ⊢ Ⅎ𝑦[⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑
33		nfcv 2908	. . 3 ⊢ Ⅎ𝑦𝑉
34	32, 33	nfrabw 3483	. 2 ⊢ Ⅎ𝑦{𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}
35	2, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34	ovmpodxf 7600	1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  {crab 3443  Vcvv 3488  [wsbc 3804  ⟨cop 4654  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  1^st c1st 8028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030

This theorem is referenced by:  mpoxopovel  8261  mpoxopoveqd  8262