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Theorem sscfn2 16418
Description: The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
sscfn1.1 (𝜑𝐻cat 𝐽)
sscfn2.2 (𝜑𝑇 = dom dom 𝐽)
Assertion
Ref Expression
sscfn2 (𝜑𝐽 Fn (𝑇 × 𝑇))

Proof of Theorem sscfn2
Dummy variables 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sscfn1.1 . . 3 (𝜑𝐻cat 𝐽)
2 brssc 16414 . . 3 (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽𝑥)))
31, 2sylib 208 . 2 (𝜑 → ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽𝑥)))
4 simpr 477 . . . . . 6 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑡 × 𝑡))
5 sscfn2.2 . . . . . . . . . 10 (𝜑𝑇 = dom dom 𝐽)
65adantr 481 . . . . . . . . 9 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → 𝑇 = dom dom 𝐽)
7 fndm 5958 . . . . . . . . . . . 12 (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
87adantl 482 . . . . . . . . . . 11 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑡 × 𝑡))
98dmeqd 5296 . . . . . . . . . 10 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = dom (𝑡 × 𝑡))
10 dmxpid 5315 . . . . . . . . . 10 dom (𝑡 × 𝑡) = 𝑡
119, 10syl6eq 2671 . . . . . . . . 9 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = 𝑡)
126, 11eqtr2d 2656 . . . . . . . 8 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → 𝑡 = 𝑇)
1312sqxpeqd 5111 . . . . . . 7 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇))
1413fneq2d 5950 . . . . . 6 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑇 × 𝑇)))
154, 14mpbid 222 . . . . 5 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑇 × 𝑇))
1615ex 450 . . . 4 (𝜑 → (𝐽 Fn (𝑡 × 𝑡) → 𝐽 Fn (𝑇 × 𝑇)))
1716adantrd 484 . . 3 (𝜑 → ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽𝑥)) → 𝐽 Fn (𝑇 × 𝑇)))
1817exlimdv 1858 . 2 (𝜑 → (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽𝑥)) → 𝐽 Fn (𝑇 × 𝑇)))
193, 18mpd 15 1 (𝜑𝐽 Fn (𝑇 × 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2909  𝒫 cpw 4136   class class class wbr 4623   × cxp 5082  dom cdm 5084   Fn wfn 5852  cfv 5857  Xcixp 7868  cat cssc 16407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ixp 7869  df-ssc 16410
This theorem is referenced by:  ssc2  16422  ssctr  16425
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