Theorem mpt2xopovel 5797

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 5795	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉))
3	2	pm4.71rd 374	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾))))
4	1	mpt2xopoveq 5796	. . . . . 6 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / x][𝐾 / y]φ})
5	4	eleq2d 2104	. . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ 𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / x][𝐾 / y]φ}))
6		nfcv 2175	. . . . . . 7 ⊢ Ⅎ𝑛𝑉
7	6	elrabsf 2795	. . . . . 6 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / x][𝐾 / y]φ} ↔ (𝑁 ∈ 𝑉 ∧ [𝑁 / 𝑛][⟨𝑉, 𝑊⟩ / x][𝐾 / y]φ))
8		sbccom 2827	. . . . . . . 8 ⊢ ([𝑁 / 𝑛][⟨𝑉, 𝑊⟩ / x][𝐾 / y]φ ↔ [⟨𝑉, 𝑊⟩ / x][𝑁 / 𝑛][𝐾 / y]φ)
9		sbccom 2827	. . . . . . . . 9 ⊢ ([𝑁 / 𝑛][𝐾 / y]φ ↔ [𝐾 / y][𝑁 / 𝑛]φ)
10	9	sbcbii 2812	. . . . . . . 8 ⊢ ([⟨𝑉, 𝑊⟩ / x][𝑁 / 𝑛][𝐾 / y]φ ↔ [⟨𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ)
11	8, 10	bitri 173	. . . . . . 7 ⊢ ([𝑁 / 𝑛][⟨𝑉, 𝑊⟩ / x][𝐾 / y]φ ↔ [⟨𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ)
12	11	anbi2i 430	. . . . . 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 173	. . . . 5 ⊢ (𝑁 ∈ {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / x][𝐾 / y]φ} ↔ (𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / x][𝐾 / y][𝑁 / 𝑛]φ))
14	5, 13	syl6bb 185	. . . 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 425	. . 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 888	. . 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 187	. 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 177	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 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  {crab 2304  Vcvv 2551  [wsbc 2758  ⟨cop 3370  ‘cfv 4845  (class class class)co 5455   ↦ cmpt2 5457  1^st c1st 5707

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710

This theorem is referenced by: (None)