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Theorem pmss12g 8425
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 5558 . . . . . . 7 ((𝐵𝐷𝐴𝐶) → (𝐵 × 𝐴) ⊆ (𝐷 × 𝐶))
21ancoms 462 . . . . . 6 ((𝐴𝐶𝐵𝐷) → (𝐵 × 𝐴) ⊆ (𝐷 × 𝐶))
3 sstr 3961 . . . . . . 7 ((𝑓 ⊆ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ⊆ (𝐷 × 𝐶)) → 𝑓 ⊆ (𝐷 × 𝐶))
43expcom 417 . . . . . 6 ((𝐵 × 𝐴) ⊆ (𝐷 × 𝐶) → (𝑓 ⊆ (𝐵 × 𝐴) → 𝑓 ⊆ (𝐷 × 𝐶)))
52, 4syl 17 . . . . 5 ((𝐴𝐶𝐵𝐷) → (𝑓 ⊆ (𝐵 × 𝐴) → 𝑓 ⊆ (𝐷 × 𝐶)))
65anim2d 614 . . . 4 ((𝐴𝐶𝐵𝐷) → ((Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴)) → (Fun 𝑓𝑓 ⊆ (𝐷 × 𝐶))))
76adantr 484 . . 3 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → ((Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴)) → (Fun 𝑓𝑓 ⊆ (𝐷 × 𝐶))))
8 ssexg 5214 . . . . 5 ((𝐴𝐶𝐶𝑉) → 𝐴 ∈ V)
9 ssexg 5214 . . . . 5 ((𝐵𝐷𝐷𝑊) → 𝐵 ∈ V)
10 elpmg 8414 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑓 ∈ (𝐴pm 𝐵) ↔ (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴))))
118, 9, 10syl2an 598 . . . 4 (((𝐴𝐶𝐶𝑉) ∧ (𝐵𝐷𝐷𝑊)) → (𝑓 ∈ (𝐴pm 𝐵) ↔ (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴))))
1211an4s 659 . . 3 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝑓 ∈ (𝐴pm 𝐵) ↔ (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴))))
13 elpmg 8414 . . . 4 ((𝐶𝑉𝐷𝑊) → (𝑓 ∈ (𝐶pm 𝐷) ↔ (Fun 𝑓𝑓 ⊆ (𝐷 × 𝐶))))
1413adantl 485 . . 3 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝑓 ∈ (𝐶pm 𝐷) ↔ (Fun 𝑓𝑓 ⊆ (𝐷 × 𝐶))))
157, 12, 143imtr4d 297 . 2 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝑓 ∈ (𝐴pm 𝐵) → 𝑓 ∈ (𝐶pm 𝐷)))
1615ssrdv 3959 1 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝐴pm 𝐵) ⊆ (𝐶pm 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  Vcvv 3480  wss 3919   × cxp 5541  Fun wfun 6338  (class class class)co 7146  pm cpm 8399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-pm 8401
This theorem is referenced by:  lmres  21903  dvnadd  24530  caures  35110
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