Theorem elpmg 8439

Description: The predicate "is a partial function." (Contributed by Mario Carneiro, 14-Nov-2013.)

Assertion

Ref	Expression
elpmg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝐶 ∧ 𝐶 ⊆ (𝐵 × 𝐴))))

Proof of Theorem elpmg

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmvalg 8434	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ↑pm 𝐵) = {𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑔})
2	1	eleq2d 2838	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑pm 𝐵) ↔ 𝐶 ∈ {𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑔}))
3		funeq 6361	. . . . 5 ⊢ (𝑔 = 𝐶 → (Fun 𝑔 ↔ Fun 𝐶))
4	3	elrab 3605	. . . 4 ⊢ (𝐶 ∈ {𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑔} ↔ (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ∧ Fun 𝐶))
5	2, 4	bitrdi 290	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ∧ Fun 𝐶)))
6	5	biancomd 467	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝐶 ∧ 𝐶 ∈ 𝒫 (𝐵 × 𝐴))))
7		elex 3429	. . . . 5 ⊢ (𝐶 ∈ 𝒫 (𝐵 × 𝐴) → 𝐶 ∈ V)
8	7	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ 𝒫 (𝐵 × 𝐴) → 𝐶 ∈ V))
9		xpexg 7478	. . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → (𝐵 × 𝐴) ∈ V)
10	9	ancoms 462	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 × 𝐴) ∈ V)
11		ssexg 5198	. . . . . 6 ⊢ ((𝐶 ⊆ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ∈ V) → 𝐶 ∈ V)
12	11	expcom 417	. . . . 5 ⊢ ((𝐵 × 𝐴) ∈ V → (𝐶 ⊆ (𝐵 × 𝐴) → 𝐶 ∈ V))
13	10, 12	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ⊆ (𝐵 × 𝐴) → 𝐶 ∈ V))
14		elpwg 4501	. . . . 5 ⊢ (𝐶 ∈ V → (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝐶 ⊆ (𝐵 × 𝐴)))
15	14	a1i 11	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ V → (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝐶 ⊆ (𝐵 × 𝐴))))
16	8, 13, 15	pm5.21ndd 384	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝐶 ⊆ (𝐵 × 𝐴)))
17	16	anbi2d 631	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((Fun 𝐶 ∧ 𝐶 ∈ 𝒫 (𝐵 × 𝐴)) ↔ (Fun 𝐶 ∧ 𝐶 ⊆ (𝐵 × 𝐴))))
18	6, 17	bitrd 282	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝐶 ∧ 𝐶 ⊆ (𝐵 × 𝐴))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2112  {crab 3075  Vcvv 3410   ⊆ wss 3861  𝒫 cpw 4498   × cxp 5527  Fun wfun 6335  (class class class)co 7157   ↑_pm cpm 8424

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pow 5239  ax-pr 5303  ax-un 7466

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3700  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-br 5038  df-opab 5100  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-iota 6300  df-fun 6343  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-pm 8426

This theorem is referenced by:  elpm2g  8440  pmss12g  8465  elpm  8469  pmsspw  8473  lmfss  22011  lmmbr2  23974  iscau2  23992  caussi  24012  causs  24013