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Theorem elovmporab1 7651
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. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker elovmporab1w 7650 when possible. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (New usage is discouraged.)
Hypotheses
Ref Expression
elovmporab1.o 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑥 / 𝑚𝑀𝜑})
elovmporab1.v ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 / 𝑚𝑀 ∈ V)
Assertion
Ref Expression
elovmporab1 (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 elovmporab1
StepHypRef Expression
1 elovmporab1.o . . 3 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑥 / 𝑚𝑀𝜑})
21elmpocl 7645 . 2 (𝑍 ∈ (𝑋𝑂𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
31a1i 11 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑧𝑥 / 𝑚𝑀𝜑}))
4 csbeq1 3891 . . . . . . 7 (𝑥 = 𝑋𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
54ad2antrl 725 . . . . . 6 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
6 sbceq1a 3783 . . . . . . . 8 (𝑦 = 𝑌 → (𝜑[𝑌 / 𝑦]𝜑))
7 sbceq1a 3783 . . . . . . . 8 (𝑥 = 𝑋 → ([𝑌 / 𝑦]𝜑[𝑋 / 𝑥][𝑌 / 𝑦]𝜑))
86, 7sylan9bbr 510 . . . . . . 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 3443 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → {𝑧𝑥 / 𝑚𝑀𝜑} = {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
11 eqidd 2727 . . . . 5 (((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑥 = 𝑋) → V = V)
12 simpl 482 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 ∈ V)
13 simpr 484 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑌 ∈ V)
14 elovmporab1.v . . . . . 6 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑋 / 𝑚𝑀 ∈ V)
15 rabexg 5324 . . . . . 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 2897 . . . . . . 7 𝑥𝑋
1817nfel1 2913 . . . . . 6 𝑥 𝑋 ∈ V
19 nfcv 2897 . . . . . . 7 𝑥𝑌
2019nfel1 2913 . . . . . 6 𝑥 𝑌 ∈ V
2118, 20nfan 1894 . . . . 5 𝑥(𝑋 ∈ V ∧ 𝑌 ∈ V)
22 nfcv 2897 . . . . . . 7 𝑦𝑋
2322nfel1 2913 . . . . . 6 𝑦 𝑋 ∈ V
24 nfcv 2897 . . . . . . 7 𝑦𝑌
2524nfel1 2913 . . . . . 6 𝑦 𝑌 ∈ V
2623, 25nfan 1894 . . . . 5 𝑦(𝑋 ∈ V ∧ 𝑌 ∈ V)
27 nfsbc1v 3792 . . . . . 6 𝑥[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
28 nfcv 2897 . . . . . . 7 𝑥𝑀
2917, 28nfcsb 3916 . . . . . 6 𝑥𝑋 / 𝑚𝑀
3027, 29nfrab 3466 . . . . 5 𝑥{𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
31 nfsbc1v 3792 . . . . . . 7 𝑦[𝑌 / 𝑦]𝜑
3222, 31nfsbc 3797 . . . . . 6 𝑦[𝑋 / 𝑥][𝑌 / 𝑦]𝜑
33 nfcv 2897 . . . . . . 7 𝑦𝑀
3422, 33nfcsb 3916 . . . . . 6 𝑦𝑋 / 𝑚𝑀
3532, 34nfrab 3466 . . . . 5 𝑦{𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑}
363, 10, 11, 12, 13, 16, 21, 26, 22, 19, 30, 35ovmpodxf 7554 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = {𝑧𝑋 / 𝑚𝑀[𝑋 / 𝑥][𝑌 / 𝑦]𝜑})
3736eleq2d 2813 . . 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 502 . . . 4 (𝑍𝑋 / 𝑚𝑀 → ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍𝑋 / 𝑚𝑀)))
40 elrabi 3672 . . . 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 239 . 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 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  {crab 3426  Vcvv 3468  [wsbc 3772  csb 3888  (class class class)co 7405  cmpo 7407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2365  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410
This theorem is referenced by: (None)
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