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Theorem elovmpt2rab1 7046
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 7041 . 2 (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
31a1i 11 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑥 / 𝑚𝑀𝜑}))
4 csbeq1 3677 . . . . . . 7 (𝑥 = 𝑋𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
54ad2antrl 766 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
6 sbceq1a 3587 . . . . . . . 8 (𝑦 = 𝑌 → (𝜑[𝑌 / 𝑦]𝜑))
7 sbceq1a 3587 . . . . . . . 8 (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
86, 7sylan9bbr 739 . . . . . . 7 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
98adantl 473 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
105, 9rabeqbidv 3335 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → {𝑧𝑥 / 𝑚𝑀𝜑} = {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
11 eqidd 2761 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑥 = 𝑋) → V = V)
12 simpl 474 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
13 simpr 479 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
14 elovmpt2rab1.v . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 / 𝑚𝑀 ∈ V)
15 rabexg 4963 . . . . . 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 2902 . . . . . . 7 𝑥𝑋
1817nfel1 2917 . . . . . 6 𝑥 𝑋 ∈ V
19 nfcv 2902 . . . . . . 7 𝑥𝑌
2019nfel1 2917 . . . . . 6 𝑥 𝑌 ∈ V
2118, 20nfan 1977 . . . . 5 𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
22 nfcv 2902 . . . . . . 7 𝑦𝑋
2322nfel1 2917 . . . . . 6 𝑦 𝑋 ∈ V
24 nfcv 2902 . . . . . . 7 𝑦𝑌
2524nfel1 2917 . . . . . 6 𝑦 𝑌 ∈ V
2623, 25nfan 1977 . . . . 5 𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
27 nfsbc1v 3596 . . . . . 6 𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
28 nfcv 2902 . . . . . . 7 𝑥𝑀
2917, 28nfcsb 3692 . . . . . 6 𝑥𝑋 / 𝑚𝑀
3027, 29nfrab 3262 . . . . 5 𝑥{𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
31 nfsbc1v 3596 . . . . . . 7 𝑦[𝑌 / 𝑦]𝜑
3222, 31nfsbc 3598 . . . . . 6 𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
33 nfcv 2902 . . . . . . 7 𝑦𝑀
3422, 33nfcsb 3692 . . . . . 6 𝑦𝑋 / 𝑚𝑀
3532, 34nfrab 3262 . . . . 5 𝑦{𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
363, 10, 11, 12, 13, 16, 21, 26, 22, 19, 30, 35ovmpt2dxf 6951 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
3736eleq2d 2825 . . 3 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑍 ∈ (𝑋𝑂𝑌) ↔ 𝑍 ∈ {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}))
38 df-3an 1074 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍𝑋 / 𝑚𝑀))
3938simplbi2com 658 . . . 4 (𝑍𝑋 / 𝑚𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀)))
40 elrabi 3499 . . . 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 230 . 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  [wsbc 3576  csb 3674  (class class class)co 6813  cmpt2 6815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818
This theorem is referenced by:  elovmpt2wrd  13534
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