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Theorem elovmpt2rab1 6779
 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. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Hypotheses
Ref Expression
elovmpt2rab1.o 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑥 / 𝑚𝑀𝜑})
elovmpt2rab1.v ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 / 𝑚𝑀 ∈ V)
Assertion
Ref Expression
elovmpt2rab1 (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   𝑧,𝑚
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 elovmpt2rab1
StepHypRef Expression
1 elovmpt2rab1.o . . 3 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑥 / 𝑚𝑀𝜑})
21elmpt2cl 6774 . 2 (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
31a1i 11 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑥 / 𝑚𝑀𝜑}))
4 csbeq1 3502 . . . . . . 7 (𝑥 = 𝑋𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
54ad2antrl 760 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
6 sbceq1a 3413 . . . . . . . 8 (𝑦 = 𝑌 → (𝜑[𝑌 / 𝑦]𝜑))
7 sbceq1a 3413 . . . . . . . 8 (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
86, 7sylan9bbr 733 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
98adantl 481 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
105, 9rabeqbidv 3168 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → {𝑧𝑥 / 𝑚𝑀𝜑} = {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
11 eqidd 2611 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑥 = 𝑋) → V = V)
12 simpl 472 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
13 simpr 476 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
14 elovmpt2rab1.v . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 / 𝑚𝑀 ∈ V)
15 rabexg 4739 . . . . . 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 2751 . . . . . . 7 𝑥𝑋
1817nfel1 2765 . . . . . 6 𝑥 𝑋 ∈ V
19 nfcv 2751 . . . . . . 7 𝑥𝑌
2019nfel1 2765 . . . . . 6 𝑥 𝑌 ∈ V
2118, 20nfan 1816 . . . . 5 𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
22 nfcv 2751 . . . . . . 7 𝑦𝑋
2322nfel1 2765 . . . . . 6 𝑦 𝑋 ∈ V
24 nfcv 2751 . . . . . . 7 𝑦𝑌
2524nfel1 2765 . . . . . 6 𝑦 𝑌 ∈ V
2623, 25nfan 1816 . . . . 5 𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
27 nfsbc1v 3422 . . . . . 6 𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
28 nfcv 2751 . . . . . . 7 𝑥𝑀
2917, 28nfcsb 3517 . . . . . 6 𝑥𝑋 / 𝑚𝑀
3027, 29nfrab 3100 . . . . 5 𝑥{𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
31 nfsbc1v 3422 . . . . . . 7 𝑦[𝑌 / 𝑦]𝜑
3222, 31nfsbc 3424 . . . . . 6 𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
33 nfcv 2751 . . . . . . 7 𝑦𝑀
3422, 33nfcsb 3517 . . . . . 6 𝑦𝑋 / 𝑚𝑀
3532, 34nfrab 3100 . . . . 5 𝑦{𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
363, 10, 11, 12, 13, 16, 21, 26, 22, 19, 30, 35ovmpt2dxf 6684 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
3736eleq2d 2673 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) ↔ 𝑍 ∈ {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
38 elrabi 3328 . . . . 5 (𝑍 ∈ {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → 𝑍𝑋 / 𝑚𝑀)
39 df-3an 1033 . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍𝑋 / 𝑚𝑀))
4039simplbi2com 655 . . . . 5 (𝑍𝑋 / 𝑚𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀)))
4138, 40syl 17 . . . 4 (𝑍 ∈ {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀)))
4241com12 32 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑} → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀)))
4337, 42sylbid 229 . 2 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀)))
442, 43mpcom 37 1 (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  [wsbc 3402  ⦋csb 3499  (class class class)co 6549   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  elovmpt2wrd  13202
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