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Theorem elfvmptrab1 5561
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 5208 . . . 4 Fun 𝐹
3 funrel 5186 . . . 4 (Fun 𝐹 → Rel 𝐹)
42, 3ax-mp 5 . . 3 Rel 𝐹
5 relelfvdm 5499 . . 3 ((Rel 𝐹𝑌 ∈ (𝐹𝑋)) → 𝑋 ∈ dom 𝐹)
64, 5mpan 421 . 2 (𝑌 ∈ (𝐹𝑋) → 𝑋 ∈ dom 𝐹)
71dmmptss 5081 . . . . . 6 dom 𝐹𝑉
87sseli 3124 . . . . 5 (𝑋 ∈ dom 𝐹𝑋𝑉)
9 elfvmptrab1.v . . . . . 6 (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
10 rabexg 4107 . . . . . 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 2299 . . . . . 6 𝑥𝑋
13 nfsbc1v 2955 . . . . . . 7 𝑥[𝑋 / 𝑥]𝜑
14 nfcv 2299 . . . . . . . 8 𝑥𝑀
1512, 14nfcsb 3068 . . . . . . 7 𝑥𝑋 / 𝑚𝑀
1613, 15nfrabxy 2637 . . . . . 6 𝑥{𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}
17 csbeq1 3034 . . . . . . 7 (𝑥 = 𝑋𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
18 sbceq1a 2946 . . . . . . 7 (𝑥 = 𝑋 → (𝜑[𝑋 / 𝑥]𝜑))
1917, 18rabeqbidv 2707 . . . . . 6 (𝑥 = 𝑋 → {𝑦𝑥 / 𝑚𝑀𝜑} = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
2012, 16, 19, 1fvmptf 5559 . . . . 5 ((𝑋𝑉 ∧ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} ∈ V) → (𝐹𝑋) = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
218, 11, 20syl2anc 409 . . . 4 (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
2221eleq2d 2227 . . 3 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}))
23 elrabi 2865 . . . . 5 (𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} → 𝑌𝑋 / 𝑚𝑀)
248, 23anim12i 336 . . . 4 ((𝑋 ∈ dom 𝐹𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
2524ex 114 . . 3 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} → (𝑋𝑉𝑌𝑋 / 𝑚𝑀)))
2622, 25sylbid 149 . 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 103   = wceq 1335  wcel 2128  {crab 2439  Vcvv 2712  [wsbc 2937  csb 3031  cmpt 4025  dom cdm 4585  Rel wrel 4590  Fun wfun 5163  cfv 5169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fv 5177
This theorem is referenced by:  elfvmptrab  5562
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