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Theorem fsovfvfvd 40657
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 40656 . . 3 (𝜑 → (𝐺𝐹) = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝐹𝑥)}))
81, 7syl5eq 2871 . 2 (𝜑𝐻 = (𝑦𝐵 ↦ {𝑥𝐴𝑦 ∈ (𝐹𝑥)}))
9 eleq1 2903 . . . 4 (𝑦 = 𝑌 → (𝑦 ∈ (𝐹𝑥) ↔ 𝑌 ∈ (𝐹𝑥)))
109rabbidv 3465 . . 3 (𝑦 = 𝑌 → {𝑥𝐴𝑦 ∈ (𝐹𝑥)} = {𝑥𝐴𝑌 ∈ (𝐹𝑥)})
1110adantl 485 . 2 ((𝜑𝑦 = 𝑌) → {𝑥𝐴𝑦 ∈ (𝐹𝑥)} = {𝑥𝐴𝑌 ∈ (𝐹𝑥)})
12 fsovfvfvd.y . 2 (𝜑𝑌𝐵)
13 rabexg 5221 . . 3 (𝐴𝑉 → {𝑥𝐴𝑌 ∈ (𝐹𝑥)} ∈ V)
143, 13syl 17 . 2 (𝜑 → {𝑥𝐴𝑌 ∈ (𝐹𝑥)} ∈ V)
158, 11, 12, 14fvmptd 6768 1 (𝜑 → (𝐻𝑌) = {𝑥𝐴𝑌 ∈ (𝐹𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {crab 3137  Vcvv 3480  𝒫 cpw 4522  cmpt 5133  cfv 6345  (class class class)co 7151  cmpo 7153  m cmap 8404
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pr 5318
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156
This theorem is referenced by:  ntrneiel  40731
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