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Theorem elpmg 7825
Description: The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.)
Assertion
Ref Expression
elpmg ((𝐴𝑉𝐵𝑊) → (𝐶 ∈ (𝐴pm 𝐵) ↔ (Fun 𝐶𝐶 ⊆ (𝐵 × 𝐴))))

Proof of Theorem elpmg
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 pmvalg 7820 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴pm 𝐵) = {𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑔})
21eleq2d 2684 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐶 ∈ (𝐴pm 𝐵) ↔ 𝐶 ∈ {𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑔}))
3 funeq 5872 . . . . 5 (𝑔 = 𝐶 → (Fun 𝑔 ↔ Fun 𝐶))
43elrab 3350 . . . 4 (𝐶 ∈ {𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑔} ↔ (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ∧ Fun 𝐶))
52, 4syl6bb 276 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐶 ∈ (𝐴pm 𝐵) ↔ (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ∧ Fun 𝐶)))
6 ancom 466 . . 3 ((𝐶 ∈ 𝒫 (𝐵 × 𝐴) ∧ Fun 𝐶) ↔ (Fun 𝐶𝐶 ∈ 𝒫 (𝐵 × 𝐴)))
75, 6syl6bb 276 . 2 ((𝐴𝑉𝐵𝑊) → (𝐶 ∈ (𝐴pm 𝐵) ↔ (Fun 𝐶𝐶 ∈ 𝒫 (𝐵 × 𝐴))))
8 elex 3201 . . . . 5 (𝐶 ∈ 𝒫 (𝐵 × 𝐴) → 𝐶 ∈ V)
98a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐶 ∈ 𝒫 (𝐵 × 𝐴) → 𝐶 ∈ V))
10 xpexg 6920 . . . . . 6 ((𝐵𝑊𝐴𝑉) → (𝐵 × 𝐴) ∈ V)
1110ancoms 469 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐵 × 𝐴) ∈ V)
12 ssexg 4769 . . . . . 6 ((𝐶 ⊆ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ∈ V) → 𝐶 ∈ V)
1312expcom 451 . . . . 5 ((𝐵 × 𝐴) ∈ V → (𝐶 ⊆ (𝐵 × 𝐴) → 𝐶 ∈ V))
1411, 13syl 17 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐶 ⊆ (𝐵 × 𝐴) → 𝐶 ∈ V))
15 elpwg 4143 . . . . 5 (𝐶 ∈ V → (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝐶 ⊆ (𝐵 × 𝐴)))
1615a1i 11 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐶 ∈ V → (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝐶 ⊆ (𝐵 × 𝐴))))
179, 14, 16pm5.21ndd 369 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝐶 ⊆ (𝐵 × 𝐴)))
1817anbi2d 739 . 2 ((𝐴𝑉𝐵𝑊) → ((Fun 𝐶𝐶 ∈ 𝒫 (𝐵 × 𝐴)) ↔ (Fun 𝐶𝐶 ⊆ (𝐵 × 𝐴))))
197, 18bitrd 268 1 ((𝐴𝑉𝐵𝑊) → (𝐶 ∈ (𝐴pm 𝐵) ↔ (Fun 𝐶𝐶 ⊆ (𝐵 × 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1987  {crab 2911  Vcvv 3189  wss 3559  𝒫 cpw 4135   × cxp 5077  Fun wfun 5846  (class class class)co 6610  pm cpm 7810
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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
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-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-pm 7812
This theorem is referenced by:  elpm2g  7826  pmss12g  7836  elpm  7840  pmsspw  7844  lmfss  21023  lmmbr2  22980  iscau2  22998  caussi  23018  causs  23019
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