Theorem pmss12g 8907

Description: Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)

Assertion

Ref	Expression
pmss12g	⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (𝐴 ↑pm 𝐵) ⊆ (𝐶 ↑pm 𝐷))

Proof of Theorem pmss12g

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpss12 5703	. . . . . . 7 ⊢ ((𝐵 ⊆ 𝐷 ∧ 𝐴 ⊆ 𝐶) → (𝐵 × 𝐴) ⊆ (𝐷 × 𝐶))
2	1	ancoms 458	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) → (𝐵 × 𝐴) ⊆ (𝐷 × 𝐶))
3		sstr 4003	. . . . . . 7 ⊢ ((𝑓 ⊆ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ⊆ (𝐷 × 𝐶)) → 𝑓 ⊆ (𝐷 × 𝐶))
4	3	expcom 413	. . . . . 6 ⊢ ((𝐵 × 𝐴) ⊆ (𝐷 × 𝐶) → (𝑓 ⊆ (𝐵 × 𝐴) → 𝑓 ⊆ (𝐷 × 𝐶)))
5	2, 4	syl 17	. . . . 5 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) → (𝑓 ⊆ (𝐵 × 𝐴) → 𝑓 ⊆ (𝐷 × 𝐶)))
6	5	anim2d 612	. . . 4 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) → ((Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴)) → (Fun 𝑓 ∧ 𝑓 ⊆ (𝐷 × 𝐶))))
7	6	adantr 480	. . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ((Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴)) → (Fun 𝑓 ∧ 𝑓 ⊆ (𝐷 × 𝐶))))
8		ssexg 5328	. . . . 5 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ 𝑉) → 𝐴 ∈ V)
9		ssexg 5328	. . . . 5 ⊢ ((𝐵 ⊆ 𝐷 ∧ 𝐷 ∈ 𝑊) → 𝐵 ∈ V)
10		elpmg 8881	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑓 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴))))
11	8, 9, 10	syl2an 596	. . . 4 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ∈ 𝑉) ∧ (𝐵 ⊆ 𝐷 ∧ 𝐷 ∈ 𝑊)) → (𝑓 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴))))
12	11	an4s 660	. . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (𝑓 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐵 × 𝐴))))
13		elpmg 8881	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (𝑓 ∈ (𝐶 ↑pm 𝐷) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐷 × 𝐶))))
14	13	adantl 481	. . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (𝑓 ∈ (𝐶 ↑pm 𝐷) ↔ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐷 × 𝐶))))
15	7, 12, 14	3imtr4d 294	. 2 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (𝑓 ∈ (𝐴 ↑pm 𝐵) → 𝑓 ∈ (𝐶 ↑pm 𝐷)))
16	15	ssrdv 4000	1 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (𝐴 ↑pm 𝐵) ⊆ (𝐶 ↑pm 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2105  Vcvv 3477   ⊆ wss 3962   × cxp 5686  Fun wfun 6556  (class class class)co 7430   ↑_pm cpm 8865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-pm 8867

This theorem is referenced by:  lmres  23323  dvnadd  25979  caures  37746