Theorem mpt2xopoveqd 7234

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
mpt2xopoveq.f	⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})
mpt2xopoveqd.1	⊢ (𝜓 → (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌))
mpt2xopoveqd.2	⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} = ∅)

Assertion

Ref	Expression
mpt2xopoveqd	⊢ (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 mpt2xopoveqd

Step	Hyp	Ref	Expression
1		mpt2xopoveqd.1	. . . 4 ⊢ (𝜓 → (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌))
2		mpt2xopoveq.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})
3	2	mpt2xopoveq 7232	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
4	3	ex 449	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐾 ∈ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
5	1, 4	syl 17	. . 3 ⊢ (𝜓 → (𝐾 ∈ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
6	5	com12 32	. 2 ⊢ (𝐾 ∈ 𝑉 → (𝜓 → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
7		df-nel 2783	. . . . . 6 ⊢ (𝐾 ∉ 𝑉 ↔ ¬ 𝐾 ∈ 𝑉)
8	2	mpt2xopynvov0 7231	. . . . . 6 ⊢ (𝐾 ∉ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)
9	7, 8	sylbir 224	. . . . 5 ⊢ (¬ 𝐾 ∈ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)
10	9	adantr 480	. . . 4 ⊢ ((¬ 𝐾 ∈ 𝑉 ∧ 𝜓) → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)
11		mpt2xopoveqd.2	. . . . . 6 ⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} = ∅)
12	11	eqcomd 2616	. . . . 5 ⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → ∅ = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
13	12	ancoms 468	. . . 4 ⊢ ((¬ 𝐾 ∈ 𝑉 ∧ 𝜓) → ∅ = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
14	10, 13	eqtrd 2644	. . 3 ⊢ ((¬ 𝐾 ∈ 𝑉 ∧ 𝜓) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
15	14	ex 449	. 2 ⊢ (¬ 𝐾 ∈ 𝑉 → (𝜓 → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
16	6, 15	pm2.61i 175	1 ⊢ (𝜓 → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  {crab 2900  Vcvv 3173  [wsbc 3402  ∅c0 3874  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060

This theorem is referenced by: (None)