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Theorem elovmpt2rab 7157
 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
elovmpt2rab.o 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑀𝜑})
elovmpt2rab.v ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑀 ∈ V)
Assertion
Ref Expression
elovmpt2rab (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 elovmpt2rab
StepHypRef Expression
1 elovmpt2rab.o . . 3 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑀𝜑})
21elmpt2cl 7153 . 2 (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
31a1i 11 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑀𝜑}))
4 sbceq1a 3663 . . . . . . . 8 (𝑦 = 𝑌 → (𝜑[𝑌 / 𝑦]𝜑))
5 sbceq1a 3663 . . . . . . . 8 (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
64, 5sylan9bbr 506 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
76adantl 475 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
87rabbidv 3386 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → {𝑧𝑀𝜑} = {𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
9 eqidd 2779 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑥 = 𝑋) → V = V)
10 simpl 476 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
11 simpr 479 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
12 elovmpt2rab.v . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑀 ∈ V)
13 rabexg 5048 . . . . . 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 2934 . . . . . . 7 𝑥𝑋
1615nfel1 2948 . . . . . 6 𝑥 𝑋 ∈ V
17 nfcv 2934 . . . . . . 7 𝑥𝑌
1817nfel1 2948 . . . . . 6 𝑥 𝑌 ∈ V
1916, 18nfan 1946 . . . . 5 𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
20 nfcv 2934 . . . . . . 7 𝑦𝑋
2120nfel1 2948 . . . . . 6 𝑦 𝑋 ∈ V
22 nfcv 2934 . . . . . . 7 𝑦𝑌
2322nfel1 2948 . . . . . 6 𝑦 𝑌 ∈ V
2421, 23nfan 1946 . . . . 5 𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
25 nfsbc1v 3672 . . . . . 6 𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
26 nfcv 2934 . . . . . 6 𝑥𝑀
2725, 26nfrab 3310 . . . . 5 𝑥{𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
28 nfsbc1v 3672 . . . . . . 7 𝑦[𝑌 / 𝑦]𝜑
2920, 28nfsbc 3674 . . . . . 6 𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
30 nfcv 2934 . . . . . 6 𝑦𝑀
3129, 30nfrab 3310 . . . . 5 𝑦{𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
323, 8, 9, 10, 11, 14, 19, 24, 20, 17, 27, 31ovmpt2dxf 7063 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
3332eleq2d 2845 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) ↔ 𝑍 ∈ {𝑧𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
34 df-3an 1073 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑀) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍𝑀))
3534simplbi2com 498 . . . 4 (𝑍𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑀)))
36 elrabi 3567 . . . 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 232 . 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 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  {crab 3094  Vcvv 3398  [wsbc 3652  (class class class)co 6922   ↦ cmpt2 6924 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927 This theorem is referenced by: (None)
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