Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fsovfvfvd Structured version   Visualization version   GIF version

Theorem fsovfvfvd 37808
Description: Value of the operator, (𝐴𝑂𝐵), which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, 𝐴 and 𝐵, when applied to function 𝐹 and element 𝑌. (Contributed by RP, 25-Apr-2021.)
Hypotheses
Ref Expression
fsovd.fs 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
fsovd.a (𝜑𝐴𝑉)
fsovd.b (𝜑𝐵𝑊)
fsovfvd.g 𝐺 = (𝐴𝑂𝐵)
fsovfvd.f (𝜑𝐹 ∈ (𝒫 𝐵𝑚 𝐴))
fsovfvfvd.h 𝐻 = (𝐺𝐹)
fsovfvfvd.y (𝜑𝑌𝐵)
Assertion
Ref Expression
fsovfvfvd (𝜑 → (𝐻𝑌) = {𝑥𝐴𝑌 ∈ (𝐹𝑥)})
Distinct variable groups:   𝐴,𝑎,𝑏,𝑓,𝑥,𝑦   𝐵,𝑎,𝑏,𝑓,𝑦   𝑓,𝐹,𝑥,𝑦   𝑥,𝑌,𝑦   𝜑,𝑎,𝑏,𝑓,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑎,𝑏)   𝐺(𝑥,𝑦,𝑓,𝑎,𝑏)   𝐻(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑂(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑊(𝑥,𝑦,𝑓,𝑎,𝑏)   𝑌(𝑓,𝑎,𝑏)

Proof of Theorem fsovfvfvd
StepHypRef Expression
1 fsovfvfvd.h . . 3 𝐻 = (𝐺𝐹)
2 fsovd.fs . . . 4 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝑎) ↦ (𝑦𝑏 ↦ {𝑥𝑎𝑦 ∈ (𝑓𝑥)})))
3 fsovd.a . . . 4 (𝜑𝐴𝑉)
4 fsovd.b . . . 4 (𝜑𝐵𝑊)
5 fsovfvd.g . . . 4 𝐺 = (𝐴𝑂𝐵)
6 fsovfvd.f . . . 4 (𝜑𝐹 ∈ (𝒫 𝐵𝑚 𝐴))
72, 3, 4, 5, 6fsovfvd 37807 . . 3 (𝜑 → (𝐺𝐹) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝐹𝑥)}))
81, 7syl5eq 2667 . 2 (𝜑𝐻 = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝐹𝑥)}))
9 eleq1 2686 . . . 4 (𝑦 = 𝑌 → (𝑦 ∈ (𝐹𝑥) ↔ 𝑌 ∈ (𝐹𝑥)))
109rabbidv 3177 . . 3 (𝑦 = 𝑌 → {𝑥𝐴𝑦 ∈ (𝐹𝑥)} = {𝑥𝐴𝑌 ∈ (𝐹𝑥)})
1110adantl 482 . 2 ((𝜑𝑦 = 𝑌) → {𝑥𝐴𝑦 ∈ (𝐹𝑥)} = {𝑥𝐴𝑌 ∈ (𝐹𝑥)})
12 fsovfvfvd.y . 2 (𝜑𝑌𝐵)
13 rabexg 4774 . . 3 (𝐴𝑉 → {𝑥𝐴𝑌 ∈ (𝐹𝑥)} ∈ V)
143, 13syl 17 . 2 (𝜑 → {𝑥𝐴𝑌 ∈ (𝐹𝑥)} ∈ V)
158, 11, 12, 14fvmptd 6247 1 (𝜑 → (𝐻𝑌) = {𝑥𝐴𝑌 ∈ (𝐹𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  𝒫 cpw 4132  cmpt 4675  cfv 5849  (class class class)co 6607  cmpt2 6609  𝑚 cmap 7805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612
This theorem is referenced by:  ntrneiel  37882
  Copyright terms: Public domain W3C validator