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Theorem elfvmptrab1 6972
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 2370. Use the weaker elfvmptrab1w 6971 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 4292 . . 3 (𝑌 ∈ (𝐹𝑋) → (𝐹𝑋) ≠ ∅)
2 ndmfv 6874 . . . 4 𝑋 ∈ dom 𝐹 → (𝐹𝑋) = ∅)
32necon1ai 2969 . . 3 ((𝐹𝑋) ≠ ∅ → 𝑋 ∈ dom 𝐹)
4 elfvmptrab1.f . . . . . . . 8 𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
54dmmptss 6191 . . . . . . 7 dom 𝐹𝑉
65sseli 3938 . . . . . 6 (𝑋 ∈ dom 𝐹𝑋𝑉)
7 elfvmptrab1.v . . . . . . 7 (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
8 rabexg 5286 . . . . . . 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 2905 . . . . . . 7 𝑥𝑋
11 nfsbc1v 3757 . . . . . . . 8 𝑥[𝑋 / 𝑥]𝜑
12 nfcv 2905 . . . . . . . . 9 𝑥𝑀
1310, 12nfcsb 3881 . . . . . . . 8 𝑥𝑋 / 𝑚𝑀
1411, 13nfrab 3441 . . . . . . 7 𝑥{𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}
15 csbeq1 3856 . . . . . . . 8 (𝑥 = 𝑋𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
16 sbceq1a 3748 . . . . . . . 8 (𝑥 = 𝑋 → (𝜑[𝑋 / 𝑥]𝜑))
1715, 16rabeqbidv 3422 . . . . . . 7 (𝑥 = 𝑋 → {𝑦𝑥 / 𝑚𝑀𝜑} = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
1810, 14, 17, 4fvmptf 6966 . . . . . 6 ((𝑋𝑉 ∧ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} ∈ V) → (𝐹𝑋) = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
196, 9, 18syl2anc 584 . . . . 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 3637 . . . . . 6 (𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} → 𝑌𝑋 / 𝑚𝑀)
226, 21anim12i 613 . . . . 5 ((𝑋 ∈ dom 𝐹𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
2322ex 413 . . . 4 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} → (𝑋𝑉𝑌𝑋 / 𝑚𝑀)))
2420, 23sylbid 239 . . 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 396   = wceq 1541  wcel 2106  wne 2941  {crab 3405  Vcvv 3443  [wsbc 3737  csb 3853  c0 4280  cmpt 5186  dom cdm 5631  cfv 6493
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2370  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fv 6501
This theorem is referenced by: (None)
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