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Theorem elfvmptrab1 6845
Description: Implications for the value of a function 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 argument of the function. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker elfvmptrab1w 6844 when possible. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (New usage is discouraged.)
Hypotheses
Ref Expression
elfvmptrab1.f 𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
elfvmptrab1.v (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
Assertion
Ref Expression
elfvmptrab1 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
Distinct variable groups:   𝑥,𝑀,𝑦   𝑥,𝑉   𝑥,𝑋,𝑦   𝑦,𝑌   𝑦,𝑚
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚)   𝐹(𝑥,𝑦,𝑚)   𝑀(𝑚)   𝑉(𝑦,𝑚)   𝑋(𝑚)   𝑌(𝑥,𝑚)

Proof of Theorem elfvmptrab1
StepHypRef Expression
1 ne0i 4249 . . 3 (𝑌 ∈ (𝐹𝑋) → (𝐹𝑋) ≠ ∅)
2 ndmfv 6747 . . . 4 𝑋 ∈ dom 𝐹 → (𝐹𝑋) = ∅)
32necon1ai 2968 . . 3 ((𝐹𝑋) ≠ ∅ → 𝑋 ∈ dom 𝐹)
4 elfvmptrab1.f . . . . . . . 8 𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
54dmmptss 6104 . . . . . . 7 dom 𝐹𝑉
65sseli 3896 . . . . . 6 (𝑋 ∈ dom 𝐹𝑋𝑉)
7 elfvmptrab1.v . . . . . . 7 (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
8 rabexg 5224 . . . . . . 7 (𝑋 / 𝑚𝑀 ∈ V → {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} ∈ V)
96, 7, 83syl 18 . . . . . 6 (𝑋 ∈ dom 𝐹 → {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} ∈ V)
10 nfcv 2904 . . . . . . 7 𝑥𝑋
11 nfsbc1v 3714 . . . . . . . 8 𝑥[𝑋 / 𝑥]𝜑
12 nfcv 2904 . . . . . . . . 9 𝑥𝑀
1310, 12nfcsb 3839 . . . . . . . 8 𝑥𝑋 / 𝑚𝑀
1411, 13nfrab 3298 . . . . . . 7 𝑥{𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}
15 csbeq1 3814 . . . . . . . 8 (𝑥 = 𝑋𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
16 sbceq1a 3705 . . . . . . . 8 (𝑥 = 𝑋 → (𝜑[𝑋 / 𝑥]𝜑))
1715, 16rabeqbidv 3396 . . . . . . 7 (𝑥 = 𝑋 → {𝑦𝑥 / 𝑚𝑀𝜑} = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
1810, 14, 17, 4fvmptf 6839 . . . . . 6 ((𝑋𝑉 ∧ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} ∈ V) → (𝐹𝑋) = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
196, 9, 18syl2anc 587 . . . . 5 (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
2019eleq2d 2823 . . . 4 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}))
21 elrabi 3596 . . . . . 6 (𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} → 𝑌𝑋 / 𝑚𝑀)
226, 21anim12i 616 . . . . 5 ((𝑋 ∈ dom 𝐹𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
2322ex 416 . . . 4 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} → (𝑋𝑉𝑌𝑋 / 𝑚𝑀)))
2420, 23sylbid 243 . . 3 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀)))
251, 3, 243syl 18 . 2 (𝑌 ∈ (𝐹𝑋) → (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀)))
2625pm2.43i 52 1 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wne 2940  {crab 3065  Vcvv 3408  [wsbc 3694  csb 3811  c0 4237  cmpt 5135  dom cdm 5551  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2371  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fv 6388
This theorem is referenced by: (None)
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