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Theorem elovmporab1w 7394
 Description: Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. Here, the base set of the class abstraction depends on the first operand. Version of elovmporab1 7395 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by Gino Giotto, 26-Jan-2024.)
Hypotheses
Ref Expression
elovmporab1w.o 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑥 / 𝑚𝑀𝜑})
elovmporab1w.v ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 / 𝑚𝑀 ∈ V)
Assertion
Ref Expression
elovmporab1w (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀))
Distinct variable groups:   𝑥,𝑀,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑧,𝑍   𝑥,𝑚,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑚)   𝑀(𝑚)   𝑂(𝑥,𝑦,𝑧,𝑚)   𝑋(𝑚)   𝑌(𝑚)   𝑍(𝑥,𝑦,𝑚)

Proof of Theorem elovmporab1w
StepHypRef Expression
1 elovmporab1w.o . . 3 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑥 / 𝑚𝑀𝜑})
21elmpocl 7389 . 2 (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
31a1i 11 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑥 / 𝑚𝑀𝜑}))
4 csbeq1 3810 . . . . . . 7 (𝑥 = 𝑋𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
54ad2antrl 727 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
6 sbceq1a 3709 . . . . . . . 8 (𝑦 = 𝑌 → (𝜑[𝑌 / 𝑦]𝜑))
7 sbceq1a 3709 . . . . . . . 8 (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
86, 7sylan9bbr 514 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
98adantl 485 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
105, 9rabeqbidv 3398 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → {𝑧𝑥 / 𝑚𝑀𝜑} = {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
11 eqidd 2759 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑥 = 𝑋) → V = V)
12 simpl 486 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
13 simpr 488 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
14 elovmporab1w.v . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 / 𝑚𝑀 ∈ V)
15 rabexg 5205 . . . . . 6 (𝑋 / 𝑚𝑀 ∈ V → {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
1614, 15syl 17 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} ∈ V)
17 nfcv 2919 . . . . . . 7 𝑥𝑋
1817nfel1 2935 . . . . . 6 𝑥 𝑋 ∈ V
19 nfcv 2919 . . . . . . 7 𝑥𝑌
2019nfel1 2935 . . . . . 6 𝑥 𝑌 ∈ V
2118, 20nfan 1900 . . . . 5 𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
22 nfcv 2919 . . . . . . 7 𝑦𝑋
2322nfel1 2935 . . . . . 6 𝑦 𝑋 ∈ V
24 nfcv 2919 . . . . . . 7 𝑦𝑌
2524nfel1 2935 . . . . . 6 𝑦 𝑌 ∈ V
2623, 25nfan 1900 . . . . 5 𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
27 nfsbc1v 3718 . . . . . 6 𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
28 nfcv 2919 . . . . . . 7 𝑥𝑀
2917, 28nfcsbw 3833 . . . . . 6 𝑥𝑋 / 𝑚𝑀
3027, 29nfrabw 3303 . . . . 5 𝑥{𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
31 nfsbc1v 3718 . . . . . . 7 𝑦[𝑌 / 𝑦]𝜑
3222, 31nfsbcw 3720 . . . . . 6 𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
33 nfcv 2919 . . . . . . 7 𝑦𝑀
3422, 33nfcsbw 3833 . . . . . 6 𝑦𝑋 / 𝑚𝑀
3532, 34nfrabw 3303 . . . . 5 𝑦{𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
363, 10, 11, 12, 13, 16, 21, 26, 22, 19, 30, 35ovmpodxf 7301 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
3736eleq2d 2837 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) ↔ 𝑍 ∈ {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
38 df-3an 1086 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍𝑋 / 𝑚𝑀))
3938simplbi2com 506 . . . 4 (𝑍𝑋 / 𝑚𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀)))
40 elrabi 3598 . . . 4 (𝑍 ∈ {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → 𝑍𝑋 / 𝑚𝑀)
4139, 40syl11 33 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀)))
4237, 41sylbid 243 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀)))
432, 42mpcom 38 1 (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {crab 3074  Vcvv 3409  [wsbc 3698  ⦋csb 3807  (class class class)co 7156   ∈ cmpo 7158 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 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6299  df-fun 6342  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161 This theorem is referenced by:  elovmpowrd  13970
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