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Theorem elovmporab 7371
Description: Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Hypotheses
Ref Expression
elovmporab.o 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑀𝜑})
elovmporab.v ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑀 ∈ V)
Assertion
Ref Expression
elovmporab (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑀))
Distinct variable groups:   𝑥,𝑀,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑧,𝑍
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝑂(𝑥,𝑦,𝑧)   𝑍(𝑥,𝑦)

Proof of Theorem elovmporab
StepHypRef Expression
1 elovmporab.o . . 3 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑀𝜑})
21elmpocl 7367 . 2 (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
31a1i 11 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑀𝜑}))
4 sbceq1a 3731 . . . . . . . 8 (𝑦 = 𝑌 → (𝜑[𝑌 / 𝑦]𝜑))
5 sbceq1a 3731 . . . . . . . 8 (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
64, 5sylan9bbr 514 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
76adantl 485 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
87rabbidv 3427 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → {𝑧𝑀𝜑} = {𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
9 eqidd 2799 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑥 = 𝑋) → V = V)
10 simpl 486 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
11 simpr 488 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
12 elovmporab.v . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑀 ∈ V)
13 rabexg 5198 . . . . . 6 (𝑀 ∈ V → {𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
1412, 13syl 17 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
15 nfcv 2955 . . . . . . 7 𝑥𝑋
1615nfel1 2971 . . . . . 6 𝑥 𝑋 ∈ V
17 nfcv 2955 . . . . . . 7 𝑥𝑌
1817nfel1 2971 . . . . . 6 𝑥 𝑌 ∈ V
1916, 18nfan 1900 . . . . 5 𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
20 nfcv 2955 . . . . . . 7 𝑦𝑋
2120nfel1 2971 . . . . . 6 𝑦 𝑋 ∈ V
22 nfcv 2955 . . . . . . 7 𝑦𝑌
2322nfel1 2971 . . . . . 6 𝑦 𝑌 ∈ V
2421, 23nfan 1900 . . . . 5 𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
25 nfsbc1v 3740 . . . . . 6 𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
26 nfcv 2955 . . . . . 6 𝑥𝑀
2725, 26nfrabw 3338 . . . . 5 𝑥{𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
28 nfsbc1v 3740 . . . . . . 7 𝑦[𝑌 / 𝑦]𝜑
2920, 28nfsbcw 3742 . . . . . 6 𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
30 nfcv 2955 . . . . . 6 𝑦𝑀
3129, 30nfrabw 3338 . . . . 5 𝑦{𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
323, 8, 9, 10, 11, 14, 19, 24, 20, 17, 27, 31ovmpodxf 7279 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
3332eleq2d 2875 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) ↔ 𝑍 ∈ {𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
34 df-3an 1086 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑀) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍𝑀))
3534simplbi2com 506 . . . 4 (𝑍𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑀)))
36 elrabi 3623 . . . 4 (𝑍 ∈ {𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → 𝑍𝑀)
3735, 36syl11 33 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ {𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑀)))
3833, 37sylbid 243 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑀)))
392, 38mpcom 38 1 (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  {crab 3110  Vcvv 3441  [wsbc 3720  (class class class)co 7135  cmpo 7137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140
This theorem is referenced by: (None)
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