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Theorem pmvalg 8779
Description: The value of the partial mapping operation. (𝐴pm 𝐵) is the set of all partial functions that map from 𝐵 to 𝐴. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
pmvalg ((𝐴𝐶𝐵𝐷) → (𝐴pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓
Allowed substitution hints:   𝐶(𝑓)   𝐷(𝑓)

Proof of Theorem pmvalg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 4038 . . 3 {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ⊆ 𝒫 (𝐵 × 𝐴)
2 xpexg 7685 . . . . 5 ((𝐵𝐷𝐴𝐶) → (𝐵 × 𝐴) ∈ V)
32ancoms 460 . . . 4 ((𝐴𝐶𝐵𝐷) → (𝐵 × 𝐴) ∈ V)
43pwexd 5335 . . 3 ((𝐴𝐶𝐵𝐷) → 𝒫 (𝐵 × 𝐴) ∈ V)
5 ssexg 5281 . . 3 (({𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ⊆ 𝒫 (𝐵 × 𝐴) ∧ 𝒫 (𝐵 × 𝐴) ∈ V) → {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V)
61, 4, 5sylancr 588 . 2 ((𝐴𝐶𝐵𝐷) → {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V)
7 elex 3462 . . 3 (𝐴𝐶𝐴 ∈ V)
8 elex 3462 . . 3 (𝐵𝐷𝐵 ∈ V)
9 xpeq2 5655 . . . . . . 7 (𝑥 = 𝐴 → (𝑦 × 𝑥) = (𝑦 × 𝐴))
109pweqd 4578 . . . . . 6 (𝑥 = 𝐴 → 𝒫 (𝑦 × 𝑥) = 𝒫 (𝑦 × 𝐴))
1110rabeqdv 3421 . . . . 5 (𝑥 = 𝐴 → {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓})
12 xpeq1 5648 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 × 𝐴) = (𝐵 × 𝐴))
1312pweqd 4578 . . . . . 6 (𝑦 = 𝐵 → 𝒫 (𝑦 × 𝐴) = 𝒫 (𝐵 × 𝐴))
1413rabeqdv 3421 . . . . 5 (𝑦 = 𝐵 → {𝑓 ∈ 𝒫 (𝑦 × 𝐴) ∣ Fun 𝑓} = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
15 df-pm 8771 . . . . 5 pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
1611, 14, 15ovmpog 7515 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V) → (𝐴pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
17163expia 1122 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V → (𝐴pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓}))
187, 8, 17syl2an 597 . 2 ((𝐴𝐶𝐵𝐷) → ({𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓} ∈ V → (𝐴pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓}))
196, 18mpd 15 1 ((𝐴𝐶𝐵𝐷) → (𝐴pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {crab 3406  Vcvv 3444  wss 3911  𝒫 cpw 4561   × cxp 5632  Fun wfun 6491  (class class class)co 7358  pm cpm 8769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-pm 8771
This theorem is referenced by:  elpmg  8784
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