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Theorem pmss12g 7770
 Description: Subset relation for the set of partial functions. (Contributed by Mario Carneiro, 31-Dec-2013.)
Assertion
Ref Expression
pmss12g (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝐴pm 𝐵) ⊆ (𝐶pm 𝐷))

Proof of Theorem pmss12g
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 xpss12 5148 . . . . . . 7 ((𝐵𝐷𝐴𝐶) → (𝐵 × 𝐴) ⊆ (𝐷 × 𝐶))
21ancoms 468 . . . . . 6 ((𝐴𝐶𝐵𝐷) → (𝐵 × 𝐴) ⊆ (𝐷 × 𝐶))
3 sstr 3576 . . . . . . 7 ((𝑓 ⊆ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ⊆ (𝐷 × 𝐶)) → 𝑓 ⊆ (𝐷 × 𝐶))
43expcom 450 . . . . . 6 ((𝐵 × 𝐴) ⊆ (𝐷 × 𝐶) → (𝑓 ⊆ (𝐵 × 𝐴) → 𝑓 ⊆ (𝐷 × 𝐶)))
52, 4syl 17 . . . . 5 ((𝐴𝐶𝐵𝐷) → (𝑓 ⊆ (𝐵 × 𝐴) → 𝑓 ⊆ (𝐷 × 𝐶)))
65anim2d 587 . . . 4 ((𝐴𝐶𝐵𝐷) → ((Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴)) → (Fun 𝑓𝑓 ⊆ (𝐷 × 𝐶))))
76adantr 480 . . 3 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → ((Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴)) → (Fun 𝑓𝑓 ⊆ (𝐷 × 𝐶))))
8 ssexg 4732 . . . . 5 ((𝐴𝐶𝐶𝑉) → 𝐴 ∈ V)
9 ssexg 4732 . . . . 5 ((𝐵𝐷𝐷𝑊) → 𝐵 ∈ V)
10 elpmg 7759 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑓 ∈ (𝐴pm 𝐵) ↔ (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴))))
118, 9, 10syl2an 493 . . . 4 (((𝐴𝐶𝐶𝑉) ∧ (𝐵𝐷𝐷𝑊)) → (𝑓 ∈ (𝐴pm 𝐵) ↔ (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴))))
1211an4s 865 . . 3 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝑓 ∈ (𝐴pm 𝐵) ↔ (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴))))
13 elpmg 7759 . . . 4 ((𝐶𝑉𝐷𝑊) → (𝑓 ∈ (𝐶pm 𝐷) ↔ (Fun 𝑓𝑓 ⊆ (𝐷 × 𝐶))))
1413adantl 481 . . 3 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝑓 ∈ (𝐶pm 𝐷) ↔ (Fun 𝑓𝑓 ⊆ (𝐷 × 𝐶))))
157, 12, 143imtr4d 282 . 2 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝑓 ∈ (𝐴pm 𝐵) → 𝑓 ∈ (𝐶pm 𝐷)))
1615ssrdv 3574 1 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝐴pm 𝐵) ⊆ (𝐶pm 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540   × cxp 5036  Fun wfun 5798  (class class class)co 6549   ↑pm cpm 7745 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pm 7747 This theorem is referenced by:  lmres  20914  dvnadd  23498  caures  32726
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