ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elfvmptrab1 GIF version

Theorem elfvmptrab1 5652
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. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
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 elfvmptrab1.f . . . . 5 𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
21funmpt2 5293 . . . 4 Fun 𝐹
3 funrel 5271 . . . 4 (Fun 𝐹 → Rel 𝐹)
42, 3ax-mp 5 . . 3 Rel 𝐹
5 relelfvdm 5586 . . 3 ((Rel 𝐹𝑌 ∈ (𝐹𝑋)) → 𝑋 ∈ dom 𝐹)
64, 5mpan 424 . 2 (𝑌 ∈ (𝐹𝑋) → 𝑋 ∈ dom 𝐹)
71dmmptss 5162 . . . . . 6 dom 𝐹𝑉
87sseli 3175 . . . . 5 (𝑋 ∈ dom 𝐹𝑋𝑉)
9 elfvmptrab1.v . . . . . 6 (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
10 rabexg 4172 . . . . . 6 (𝑋 / 𝑚𝑀 ∈ V → {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} ∈ V)
118, 9, 103syl 17 . . . . 5 (𝑋 ∈ dom 𝐹 → {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} ∈ V)
12 nfcv 2336 . . . . . 6 𝑥𝑋
13 nfsbc1v 3004 . . . . . . 7 𝑥[𝑋 / 𝑥]𝜑
14 nfcv 2336 . . . . . . . 8 𝑥𝑀
1512, 14nfcsb 3118 . . . . . . 7 𝑥𝑋 / 𝑚𝑀
1613, 15nfrabw 2675 . . . . . 6 𝑥{𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}
17 csbeq1 3083 . . . . . . 7 (𝑥 = 𝑋𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
18 sbceq1a 2995 . . . . . . 7 (𝑥 = 𝑋 → (𝜑[𝑋 / 𝑥]𝜑))
1917, 18rabeqbidv 2755 . . . . . 6 (𝑥 = 𝑋 → {𝑦𝑥 / 𝑚𝑀𝜑} = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
2012, 16, 19, 1fvmptf 5650 . . . . 5 ((𝑋𝑉 ∧ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} ∈ V) → (𝐹𝑋) = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
218, 11, 20syl2anc 411 . . . 4 (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
2221eleq2d 2263 . . 3 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}))
23 elrabi 2913 . . . . 5 (𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} → 𝑌𝑋 / 𝑚𝑀)
248, 23anim12i 338 . . . 4 ((𝑋 ∈ dom 𝐹𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
2524ex 115 . . 3 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} → (𝑋𝑉𝑌𝑋 / 𝑚𝑀)))
2622, 25sylbid 150 . 2 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀)))
276, 26mpcom 36 1 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2164  {crab 2476  Vcvv 2760  [wsbc 2985  csb 3080  cmpt 4090  dom cdm 4659  Rel wrel 4664  Fun wfun 5248  cfv 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fv 5262
This theorem is referenced by:  elfvmptrab  5653
  Copyright terms: Public domain W3C validator