Theorem pmsspw 8822

Description: Partial maps are a subset of the power set of the Cartesian product of its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)

Assertion

Ref	Expression
pmsspw	⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)

Proof of Theorem pmsspw

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0i 4275	. . . . . . 7 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → ¬ (𝐴 ↑pm 𝐵) = ∅)
2		fnpm 8778	. . . . . . . . 9 ⊢ ↑pm Fn (V × V)
3	2	fndmi 6596	. . . . . . . 8 ⊢ dom ↑pm = (V × V)
4	3	ndmov 7547	. . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑pm 𝐵) = ∅)
5	1, 4	nsyl2 141	. . . . . 6 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
6		elpmg 8787	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑓 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴))))
7	5, 6	syl 17	. . . . 5 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (𝑓 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴))))
8	7	ibi 268	. . . 4 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴)))
9	8	simprd 496	. . 3 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → 𝑓 ⊆ (𝐵 × 𝐴))
10		velpw 4541	. . 3 ⊢ (𝑓 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑓 ⊆ (𝐵 × 𝐴))
11	9, 10	sylibr 235	. 2 ⊢ (𝑓 ∈ (𝐴 ↑pm 𝐵) → 𝑓 ∈ 𝒫 (𝐵 × 𝐴))
12	11	ssriv 3926	1 ⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432   ⊆ wss 3890  ∅c0 4268  𝒫 cpw 4536   × cxp 5623  Fun wfun 6486  (class class class)co 7363   ↑_pm cpm 8771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-pm 8773

This theorem is referenced by:  mapsspw  8823  wunpm  10646