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Theorem sscfn2 16940
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 16936 . . 3 (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽𝑥)))
31, 2sylib 210 . 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 473 . . . . . . . . 9 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → 𝑇 = dom dom 𝐽)
7 fndm 6282 . . . . . . . . . . . 12 (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
87adantl 474 . . . . . . . . . . 11 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑡 × 𝑡))
98dmeqd 5618 . . . . . . . . . 10 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = dom (𝑡 × 𝑡))
10 dmxpid 5637 . . . . . . . . . 10 dom (𝑡 × 𝑡) = 𝑡
119, 10syl6eq 2824 . . . . . . . . 9 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = 𝑡)
126, 11eqtr2d 2809 . . . . . . . 8 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → 𝑡 = 𝑇)
1312sqxpeqd 5433 . . . . . . 7 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇))
1413fneq2d 6274 . . . . . 6 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑇 × 𝑇)))
154, 14mpbid 224 . . . . 5 ((𝜑𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑇 × 𝑇))
1615ex 405 . . . 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 1892 . 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 387   = wceq 1507  wex 1742  wcel 2050  wrex 3083  𝒫 cpw 4416   class class class wbr 4923   × cxp 5399  dom cdm 5401   Fn wfn 6177  cfv 6182  Xcixp 8253  cat cssc 16929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ixp 8254  df-ssc 16932
This theorem is referenced by:  ssc2  16944  ssctr  16947
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