Theorem mpoxopoveqd 8226

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, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.)

Hypotheses

Ref	Expression
mpoxopoveq.f	⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})
mpoxopoveqd.1	⊢ (𝜓 → (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌))
mpoxopoveqd.2	⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} = ∅)

Assertion

Ref	Expression
mpoxopoveqd	⊢ (𝜓 → (⟨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   𝑥,𝐹

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

Proof of Theorem mpoxopoveqd

Step	Hyp	Ref	Expression
1		mpoxopoveq.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})
2	1	mpoxopoveq 8224	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
3	2	ex 418	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐾 ∈ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
4		mpoxopoveqd.1	. . 3 ⊢ (𝜓 → (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌))
5	3, 4	syl11 34	. 2 ⊢ (𝐾 ∈ 𝑉 → (𝜓 → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
6		df-nel 3068	. . . . . 6 ⊢ (𝐾 ∉ 𝑉 ↔ ¬ 𝐾 ∈ 𝑉)
7	1	mpoxopynvov0 8223	. . . . . 6 ⊢ (𝐾 ∉ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)
8	6, 7	sylbir 238	. . . . 5 ⊢ (¬ 𝐾 ∈ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)
9	8	adantr 486	. . . 4 ⊢ ((¬ 𝐾 ∈ 𝑉 ∧ 𝜓) → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)
10		mpoxopoveqd.2	. . . . . 6 ⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} = ∅)
11	10	eqcomd 2772	. . . . 5 ⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → ∅ = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
12	11	ancoms 464	. . . 4 ⊢ ((¬ 𝐾 ∈ 𝑉 ∧ 𝜓) → ∅ = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
13	9, 12	eqtrd 2801	. . 3 ⊢ ((¬ 𝐾 ∈ 𝑉 ∧ 𝜓) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
14	13	ex 418	. 2 ⊢ (¬ 𝐾 ∈ 𝑉 → (𝜓 → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
15	5, 14	pm2.61i 184	1 ⊢ (𝜓 → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ∉ wnel 3067  {crab 3419  Vcvv 3458  [wsbc 3747  ∅c0 4289  ⟨cop 4600  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425  1^st c1st 7993

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996

This theorem is used by: (None)