Theorem mpoxopovel 8221

Description: Element of the 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 and Mario Carneiro, 10-Oct-2017.)

Hypothesis

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

Assertion

Ref	Expression
mpoxopovel	⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))

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

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

Proof of Theorem mpoxopovel

Step	Hyp	Ref	Expression
1		mpoxopoveq.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})
2	1	mpoxopn0yelv 8214	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉))
3	2	pm4.71rd 572	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾))))
4	1	mpoxopoveq 8220	. . . . . 6 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
5	4	eleq2d 2848	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
6		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑛𝑉
7	6	elrabsf 3787	. . . . . 6 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁 ∈ 𝑉 ∧ [𝑁 / 𝑛][⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
8		sbccom 3821	. . . . . . . 8 ⊢ ([𝑁 / 𝑛][⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑)
9		sbccom 3821	. . . . . . . . 9 ⊢ ([𝑁 / 𝑛][𝐾 / 𝑦]𝜑 ↔ [𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
10	9	sbcbii 3798	. . . . . . . 8 ⊢ ([⟨𝑉, 𝑊⟩ / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
11	8, 10	bitri 278	. . . . . . 7 ⊢ ([𝑁 / 𝑛][⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
12	11	anbi2i 635	. . . . . 6 ⊢ ((𝑁 ∈ 𝑉 ∧ [𝑁 / 𝑛][⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑) ↔ (𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))
13	7, 12	bitri 278	. . . . 5 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))
14	5, 13	bitrdi 290	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
15	14	pm5.32da 590	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾)) ↔ (𝐾 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))))
16		3anass 1111	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑) ↔ (𝐾 ∈ 𝑉 ∧ (𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
17	15, 16	bitr4di 292	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → ((𝐾 ∈ 𝑉 ∧ 𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾)) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
18	3, 17	bitrd 282	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453  [wsbc 3742  ⟨cop 4593  ‘cfv 6537  (class class class)co 7416   ∈ cmpo 7418  1^st c1st 7987

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990

This theorem is used by: (None)