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Theorem fsovfvfvd 39145
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 39144 . . 3 (𝜑 → (𝐺𝐹) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝐹𝑥)}))
81, 7syl5eq 2873 . 2 (𝜑𝐻 = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝐹𝑥)}))
9 eleq1 2894 . . . 4 (𝑦 = 𝑌 → (𝑦 ∈ (𝐹𝑥) ↔ 𝑌 ∈ (𝐹𝑥)))
109rabbidv 3402 . . 3 (𝑦 = 𝑌 → {𝑥𝐴𝑦 ∈ (𝐹𝑥)} = {𝑥𝐴𝑌 ∈ (𝐹𝑥)})
1110adantl 475 . 2 ((𝜑𝑦 = 𝑌) → {𝑥𝐴𝑦 ∈ (𝐹𝑥)} = {𝑥𝐴𝑌 ∈ (𝐹𝑥)})
12 fsovfvfvd.y . 2 (𝜑𝑌𝐵)
13 rabexg 5036 . . 3 (𝐴𝑉 → {𝑥𝐴𝑌 ∈ (𝐹𝑥)} ∈ V)
143, 13syl 17 . 2 (𝜑 → {𝑥𝐴𝑌 ∈ (𝐹𝑥)} ∈ V)
158, 11, 12, 14fvmptd 6535 1 (𝜑 → (𝐻𝑌) = {𝑥𝐴𝑌 ∈ (𝐹𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1658  wcel 2166  {crab 3121  Vcvv 3414  𝒫 cpw 4378  cmpt 4952  cfv 6123  (class class class)co 6905  cmpt2 6907  𝑚 cmap 8122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910
This theorem is referenced by:  ntrneiel  39219
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