Theorem mpt2xopovel 5912

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
mpt2xopoveq.f	⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})

Assertion

Ref	Expression
mpt2xopovel	⊢ ((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 mpt2xopovel

Step	Hyp	Ref	Expression
1		mpt2xopoveq.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑})
2	1	mpt2xopn0yelv 5910	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉))
3	2	pm4.71rd 386	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾))))
4	1	mpt2xopoveq 5911	. . . . . 6 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
5	4	eleq2d 2152	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
6		nfcv 2223	. . . . . . 7 ⊢ Ⅎ𝑛𝑉
7	6	elrabsf 2862	. . . . . 6 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁 ∈ 𝑉 ∧ [𝑁 / 𝑛][⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
8		sbccom 2899	. . . . . . . 8 ⊢ ([𝑁 / 𝑛][⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑)
9		sbccom 2899	. . . . . . . . 9 ⊢ ([𝑁 / 𝑛][𝐾 / 𝑦]𝜑 ↔ [𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
10	9	sbcbii 2883	. . . . . . . 8 ⊢ ([⟨𝑉, 𝑊⟩ / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
11	8, 10	bitri 182	. . . . . . 7 ⊢ ([𝑁 / 𝑛][⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑 ↔ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
12	11	anbi2i 445	. . . . . 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 182	. . . . 5 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))
14	5, 13	syl6bb 194	. . . 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 440	. . 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 924	. . 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	syl6bbr 196	. 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 186	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 102   ↔ wb 103   ∧ w3a 920   = wceq 1285   ∈ wcel 1434  {crab 2357  Vcvv 2611  [wsbc 2825  ⟨cop 3420  ‘cfv 4953  (class class class)co 5565   ↦ cmpt2 5567  1^st c1st 5818

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2613  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fv 4961  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-1st 5820  df-2nd 5821

This theorem is referenced by: (None)