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Theorem mpoxopovel 8155
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
mpoxopoveq.f 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})
Assertion
Ref Expression
mpoxopovel ((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 mpoxopovel
StepHypRef Expression
1 mpoxopoveq.f . . . 4 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st𝑥) ↦ {𝑛 ∈ (1st𝑥) ∣ 𝜑})
21mpoxopn0yelv 8148 . . 3 ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → 𝐾𝑉))
32pm4.71rd 564 . 2 ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝐾𝑉𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾))))
41mpoxopoveq 8154 . . . . . 6 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
54eleq2d 2820 . . . . 5 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ 𝑁 ∈ {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
6 nfcv 2904 . . . . . . 7 𝑛𝑉
76elrabsf 3791 . . . . . 6 (𝑁 ∈ {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁𝑉[𝑁 / 𝑛][𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
8 sbccom 3831 . . . . . . . 8 ([𝑁 / 𝑛][𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑)
9 sbccom 3831 . . . . . . . . 9 ([𝑁 / 𝑛][𝐾 / 𝑦]𝜑[𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
109sbcbii 3803 . . . . . . . 8 ([𝑉, 𝑊⟩ / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
118, 10bitri 275 . . . . . . 7 ([𝑁 / 𝑛][𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
1211anbi2i 624 . . . . . 6 ((𝑁𝑉[𝑁 / 𝑛][𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑) ↔ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))
137, 12bitri 275 . . . . 5 (𝑁 ∈ {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))
145, 13bitrdi 287 . . . 4 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
1514pm5.32da 580 . . 3 ((𝑉𝑋𝑊𝑌) → ((𝐾𝑉𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾)) ↔ (𝐾𝑉 ∧ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))))
16 3anass 1096 . . 3 ((𝐾𝑉𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑) ↔ (𝐾𝑉 ∧ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
1715, 16bitr4di 289 . 2 ((𝑉𝑋𝑊𝑌) → ((𝐾𝑉𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾)) ↔ (𝐾𝑉𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
183, 17bitrd 279 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {crab 3406  Vcvv 3447  [wsbc 3743  cop 4596  cfv 6500  (class class class)co 7361  cmpo 7363  1st c1st 7923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926
This theorem is referenced by: (None)
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