MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2xopovel Structured version   Visualization version   GIF version

Theorem mpt2xopovel 7516
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 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})
Assertion
Ref Expression
mpt2xopovel ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝐾𝑉𝑁𝑉[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
StepHypRef Expression
1 mpt2xopoveq.f . . . 4 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})
21mpt2xopn0yelv 7509 . . 3 ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → 𝐾𝑉))
32pm4.71rd 670 . 2 ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝐾𝑉𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾))))
41mpt2xopoveq 7515 . . . . . 6 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
54eleq2d 2825 . . . . 5 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ 𝑁 ∈ {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
6 nfcv 2902 . . . . . . 7 𝑛𝑉
76elrabsf 3615 . . . . . 6 (𝑁 ∈ {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁𝑉[𝑁 / 𝑛][𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
8 sbccom 3650 . . . . . . . 8 ([𝑁 / 𝑛][𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑)
9 sbccom 3650 . . . . . . . . 9 ([𝑁 / 𝑛][𝐾 / 𝑦]𝜑[𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
109sbcbii 3632 . . . . . . . 8 ([𝑉, 𝑊⟩ / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
118, 10bitri 264 . . . . . . 7 ([𝑁 / 𝑛][𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
1211anbi2i 732 . . . . . 6 ((𝑁𝑉[𝑁 / 𝑛][𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑) ↔ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))
137, 12bitri 264 . . . . 5 (𝑁 ∈ {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))
145, 13syl6bb 276 . . . 4 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
1514pm5.32da 676 . . 3 ((𝑉𝑋𝑊𝑌) → ((𝐾𝑉𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾)) ↔ (𝐾𝑉 ∧ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))))
16 3anass 1081 . . 3 ((𝐾𝑉𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑) ↔ (𝐾𝑉 ∧ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
1715, 16syl6bbr 278 . 2 ((𝑉𝑋𝑊𝑌) → ((𝐾𝑉𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾)) ↔ (𝐾𝑉𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
183, 17bitrd 268 1 ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝐾𝑉𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  [wsbc 3576  cop 4327  cfv 6049  (class class class)co 6814  cmpt2 6816  1st c1st 7332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator