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Theorem elfvmptrab1 6773
 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 2379. Use the weaker elfvmptrab1w 6772 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 4250 . . 3 (𝑌 ∈ (𝐹𝑋) → (𝐹𝑋) ≠ ∅)
2 ndmfv 6676 . . . 4 𝑋 ∈ dom 𝐹 → (𝐹𝑋) = ∅)
32necon1ai 3014 . . 3 ((𝐹𝑋) ≠ ∅ → 𝑋 ∈ dom 𝐹)
4 elfvmptrab1.f . . . . . . . 8 𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
54dmmptss 6063 . . . . . . 7 dom 𝐹𝑉
65sseli 3911 . . . . . 6 (𝑋 ∈ dom 𝐹𝑋𝑉)
7 elfvmptrab1.v . . . . . . 7 (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
8 rabexg 5199 . . . . . . 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 2955 . . . . . . 7 𝑥𝑋
11 nfsbc1v 3740 . . . . . . . 8 𝑥[𝑋 / 𝑥]𝜑
12 nfcv 2955 . . . . . . . . 9 𝑥𝑀
1310, 12nfcsb 3855 . . . . . . . 8 𝑥𝑋 / 𝑚𝑀
1411, 13nfrab 3339 . . . . . . 7 𝑥{𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}
15 csbeq1 3831 . . . . . . . 8 (𝑥 = 𝑋𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
16 sbceq1a 3731 . . . . . . . 8 (𝑥 = 𝑋 → (𝜑[𝑋 / 𝑥]𝜑))
1715, 16rabeqbidv 3433 . . . . . . 7 (𝑥 = 𝑋 → {𝑦𝑥 / 𝑚𝑀𝜑} = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
1810, 14, 17, 4fvmptf 6767 . . . . . 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 2875 . . . 4 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}))
21 elrabi 3623 . . . . . 6 (𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} → 𝑌𝑋 / 𝑚𝑀)
226, 21anim12i 615 . . . . 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 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110  Vcvv 3441  [wsbc 3720  ⦋csb 3828  ∅c0 4243   ↦ cmpt 5111  dom cdm 5520  ‘cfv 6325 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fv 6333 This theorem is referenced by: (None)
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