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Theorem mpoxopovel 6220
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 6218 . . 3 ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) → 𝐾𝑉))
32pm4.71rd 392 . 2 ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝐾𝑉𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾))))
41mpoxopoveq 6219 . . . . . 6 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (⟨𝑉, 𝑊𝐹𝐾) = {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑})
54eleq2d 2240 . . . . 5 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ 𝑁 ∈ {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}))
6 nfcv 2312 . . . . . . 7 𝑛𝑉
76elrabsf 2993 . . . . . 6 (𝑁 ∈ {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁𝑉[𝑁 / 𝑛][𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑))
8 sbccom 3030 . . . . . . . 8 ([𝑁 / 𝑛][𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑)
9 sbccom 3030 . . . . . . . . 9 ([𝑁 / 𝑛][𝐾 / 𝑦]𝜑[𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
109sbcbii 3014 . . . . . . . 8 ([𝑉, 𝑊⟩ / 𝑥][𝑁 / 𝑛][𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
118, 10bitri 183 . . . . . . 7 ([𝑁 / 𝑛][𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)
1211anbi2i 454 . . . . . 6 ((𝑁𝑉[𝑁 / 𝑛][𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑) ↔ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))
137, 12bitri 183 . . . . 5 (𝑁 ∈ {𝑛𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑} ↔ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))
145, 13bitrdi 195 . . . 4 (((𝑉𝑋𝑊𝑌) ∧ 𝐾𝑉) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
1514pm5.32da 449 . . 3 ((𝑉𝑋𝑊𝑌) → ((𝐾𝑉𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾)) ↔ (𝐾𝑉 ∧ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))))
16 3anass 977 . . 3 ((𝐾𝑉𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑) ↔ (𝐾𝑉 ∧ (𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
1715, 16bitr4di 197 . 2 ((𝑉𝑋𝑊𝑌) → ((𝐾𝑉𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾)) ↔ (𝐾𝑉𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
183, 17bitrd 187 1 ((𝑉𝑋𝑊𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊𝐹𝐾) ↔ (𝐾𝑉𝑁𝑉[𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 973   = wceq 1348  wcel 2141  {crab 2452  Vcvv 2730  [wsbc 2955  cop 3586  cfv 5198  (class class class)co 5853  cmpo 5855  1st c1st 6117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120
This theorem is referenced by: (None)
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