Theorem mpoxopovel 6220

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 6218	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉))
3	2	pm4.71rd 392	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾))))
4	1	mpoxopoveq 6219	. . . . . 6 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
5	4	eleq2d 2240	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
6		nfcv 2312	. . . . . . 7 ⊢ Ⅎ𝑛𝑉
7	6	elrabsf 2993	. . . . . 6 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁 ∈ 𝑉 ∧ [𝑁 / 𝑛][⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
8		sbccom 3030	. . . . . . . 8 ⊢ ([𝑁 / 𝑛][⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑)
9		sbccom 3030	. . . . . . . . 9 ⊢ ([𝑁 / 𝑛][𝐾 / 𝑦]𝜑 ↔ [𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
10	9	sbcbii 3014	. . . . . . . 8 ⊢ ([⟨𝑉, 𝑊⟩ / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
11	8, 10	bitri 183	. . . . . . 7 ⊢ ([𝑁 / 𝑛][⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
12	11	anbi2i 454	. . . . . 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 183	. . . . 5 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))
14	5, 13	bitrdi 195	. . . 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 449	. . 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 977	. . 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 197	. 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 187	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  {crab 2452  Vcvv 2730  [wsbc 2955  ⟨cop 3586  ‘cfv 5198  (class class class)co 5853   ∈ cmpo 5855  1^st c1st 6117

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120

This theorem is referenced by: (None)