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Theorem elfvmptrab1w 7011
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. Version of elfvmptrab1 7012 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Alexander van der Vekens, 15-Jul-2018.) Avoid ax-13 2371. (Revised by Gino Giotto, 26-Jan-2024.)
Hypotheses
Ref Expression
elfvmptrab1w.f 𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
elfvmptrab1w.v (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
Assertion
Ref Expression
elfvmptrab1w (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
Distinct variable groups:   𝑥,𝑀,𝑦   𝑥,𝑉   𝑥,𝑋,𝑦   𝑦,𝑌   𝑥,𝑚,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑚)   𝐹(𝑥,𝑦,𝑚)   𝑀(𝑚)   𝑉(𝑦,𝑚)   𝑋(𝑚)   𝑌(𝑥,𝑚)

Proof of Theorem elfvmptrab1w
StepHypRef Expression
1 elfvdm 6916 . 2 (𝑌 ∈ (𝐹𝑋) → 𝑋 ∈ dom 𝐹)
2 elfvmptrab1w.f . . . . . . 7 𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
32dmmptss 6230 . . . . . 6 dom 𝐹𝑉
43sseli 3975 . . . . 5 (𝑋 ∈ dom 𝐹𝑋𝑉)
5 elfvmptrab1w.v . . . . . 6 (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
6 rabexg 5325 . . . . . 6 (𝑋 / 𝑚𝑀 ∈ V → {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} ∈ V)
74, 5, 63syl 18 . . . . 5 (𝑋 ∈ dom 𝐹 → {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} ∈ V)
8 nfcv 2903 . . . . . 6 𝑥𝑋
9 nfsbc1v 3794 . . . . . . 7 𝑥[𝑋 / 𝑥]𝜑
10 nfcv 2903 . . . . . . . 8 𝑥𝑀
118, 10nfcsbw 3917 . . . . . . 7 𝑥𝑋 / 𝑚𝑀
129, 11nfrabw 3468 . . . . . 6 𝑥{𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}
13 csbeq1 3893 . . . . . . 7 (𝑥 = 𝑋𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
14 sbceq1a 3785 . . . . . . 7 (𝑥 = 𝑋 → (𝜑[𝑋 / 𝑥]𝜑))
1513, 14rabeqbidv 3449 . . . . . 6 (𝑥 = 𝑋 → {𝑦𝑥 / 𝑚𝑀𝜑} = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
168, 12, 15, 2fvmptf 7006 . . . . 5 ((𝑋𝑉 ∧ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} ∈ V) → (𝐹𝑋) = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
174, 7, 16syl2anc 584 . . . 4 (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
1817eleq2d 2819 . . 3 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}))
19 elrabi 3674 . . . . 5 (𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} → 𝑌𝑋 / 𝑚𝑀)
204, 19anim12i 613 . . . 4 ((𝑋 ∈ dom 𝐹𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
2120ex 413 . . 3 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} → (𝑋𝑉𝑌𝑋 / 𝑚𝑀)))
2218, 21sylbid 239 . 2 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀)))
231, 22mpcom 38 1 (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {crab 3432  Vcvv 3474  [wsbc 3774  csb 3890  cmpt 5225  dom cdm 5670  cfv 6533
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-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6485  df-fun 6535  df-fv 6541
This theorem is referenced by:  elfvmptrab  7013
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