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Theorem pmss12g 8040
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 5265 . . . . . . 7 ((𝐵𝐷𝐴𝐶) → (𝐵 × 𝐴) ⊆ (𝐷 × 𝐶))
21ancoms 446 . . . . . 6 ((𝐴𝐶𝐵𝐷) → (𝐵 × 𝐴) ⊆ (𝐷 × 𝐶))
3 sstr 3760 . . . . . . 7 ((𝑓 ⊆ (𝐵 × 𝐴) ∧ (𝐵 × 𝐴) ⊆ (𝐷 × 𝐶)) → 𝑓 ⊆ (𝐷 × 𝐶))
43expcom 398 . . . . . 6 ((𝐵 × 𝐴) ⊆ (𝐷 × 𝐶) → (𝑓 ⊆ (𝐵 × 𝐴) → 𝑓 ⊆ (𝐷 × 𝐶)))
52, 4syl 17 . . . . 5 ((𝐴𝐶𝐵𝐷) → (𝑓 ⊆ (𝐵 × 𝐴) → 𝑓 ⊆ (𝐷 × 𝐶)))
65anim2d 599 . . . 4 ((𝐴𝐶𝐵𝐷) → ((Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴)) → (Fun 𝑓𝑓 ⊆ (𝐷 × 𝐶))))
76adantr 466 . . 3 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → ((Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴)) → (Fun 𝑓𝑓 ⊆ (𝐷 × 𝐶))))
8 ssexg 4939 . . . . 5 ((𝐴𝐶𝐶𝑉) → 𝐴 ∈ V)
9 ssexg 4939 . . . . 5 ((𝐵𝐷𝐷𝑊) → 𝐵 ∈ V)
10 elpmg 8029 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑓 ∈ (𝐴pm 𝐵) ↔ (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴))))
118, 9, 10syl2an 583 . . . 4 (((𝐴𝐶𝐶𝑉) ∧ (𝐵𝐷𝐷𝑊)) → (𝑓 ∈ (𝐴pm 𝐵) ↔ (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴))))
1211an4s 639 . . 3 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝑓 ∈ (𝐴pm 𝐵) ↔ (Fun 𝑓𝑓 ⊆ (𝐵 × 𝐴))))
13 elpmg 8029 . . . 4 ((𝐶𝑉𝐷𝑊) → (𝑓 ∈ (𝐶pm 𝐷) ↔ (Fun 𝑓𝑓 ⊆ (𝐷 × 𝐶))))
1413adantl 467 . . 3 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝑓 ∈ (𝐶pm 𝐷) ↔ (Fun 𝑓𝑓 ⊆ (𝐷 × 𝐶))))
157, 12, 143imtr4d 283 . 2 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝑓 ∈ (𝐴pm 𝐵) → 𝑓 ∈ (𝐶pm 𝐷)))
1615ssrdv 3758 1 (((𝐴𝐶𝐵𝐷) ∧ (𝐶𝑉𝐷𝑊)) → (𝐴pm 𝐵) ⊆ (𝐶pm 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wcel 2145  Vcvv 3351  wss 3723   × cxp 5248  Fun wfun 6024  (class class class)co 6796  pm cpm 8014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-pm 8016
This theorem is referenced by:  lmres  21325  dvnadd  23912  caures  33888
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