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Theorem elfvmptrab1 6344
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 ne0i 3954 . . 3 (𝑌 ∈ (𝐹𝑋) → (𝐹𝑋) ≠ ∅)
2 ndmfv 6256 . . . 4 𝑋 ∈ dom 𝐹 → (𝐹𝑋) = ∅)
32necon1ai 2850 . . 3 ((𝐹𝑋) ≠ ∅ → 𝑋 ∈ dom 𝐹)
4 elfvmptrab1.f . . . . . . . 8 𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})
54dmmptss 5669 . . . . . . 7 dom 𝐹𝑉
65sseli 3632 . . . . . 6 (𝑋 ∈ dom 𝐹𝑋𝑉)
7 elfvmptrab1.v . . . . . . 7 (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)
8 rabexg 4844 . . . . . . 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 2793 . . . . . . 7 𝑥𝑋
11 nfsbc1v 3488 . . . . . . . 8 𝑥[𝑋 / 𝑥]𝜑
12 nfcv 2793 . . . . . . . . 9 𝑥𝑀
1310, 12nfcsb 3584 . . . . . . . 8 𝑥𝑋 / 𝑚𝑀
1411, 13nfrab 3153 . . . . . . 7 𝑥{𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}
15 csbeq1 3569 . . . . . . . 8 (𝑥 = 𝑋𝑥 / 𝑚𝑀 = 𝑋 / 𝑚𝑀)
16 sbceq1a 3479 . . . . . . . 8 (𝑥 = 𝑋 → (𝜑[𝑋 / 𝑥]𝜑))
1715, 16rabeqbidv 3226 . . . . . . 7 (𝑥 = 𝑋 → {𝑦𝑥 / 𝑚𝑀𝜑} = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
1810, 14, 17, 4fvmptf 6340 . . . . . 6 ((𝑋𝑉 ∧ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} ∈ V) → (𝐹𝑋) = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
196, 9, 18syl2anc 694 . . . . 5 (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑})
2019eleq2d 2716 . . . 4 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ (𝐹𝑋) ↔ 𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}))
21 elrabi 3391 . . . . . 6 (𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} → 𝑌𝑋 / 𝑚𝑀)
226, 21anim12i 589 . . . . 5 ((𝑋 ∈ dom 𝐹𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑}) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
2322ex 449 . . . 4 (𝑋 ∈ dom 𝐹 → (𝑌 ∈ {𝑦𝑋 / 𝑚𝑀[𝑋 / 𝑥]𝜑} → (𝑋𝑉𝑌𝑋 / 𝑚𝑀)))
2420, 23sylbid 230 . . 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 383   = wceq 1523  wcel 2030  wne 2823  {crab 2945  Vcvv 3231  [wsbc 3468  csb 3566  c0 3948  cmpt 4762  dom cdm 5143  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  elfvmptrab  6345
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