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Theorem sscfn1 16524
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)
sscfn1.2 (𝜑𝑆 = dom dom 𝐻)
Assertion
Ref Expression
sscfn1 (𝜑𝐻 Fn (𝑆 × 𝑆))

Proof of Theorem sscfn1
Dummy variables 𝑡 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sscfn1.1 . . 3 (𝜑𝐻cat 𝐽)
2 brssc 16521 . . 3 (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
31, 2sylib 208 . 2 (𝜑 → ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
4 ixpfn 7956 . . . . . 6 (𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 Fn (𝑠 × 𝑠))
5 simpr 476 . . . . . . . 8 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑠 × 𝑠))
6 sscfn1.2 . . . . . . . . . . . 12 (𝜑𝑆 = dom dom 𝐻)
76adantr 480 . . . . . . . . . . 11 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝑆 = dom dom 𝐻)
8 fndm 6028 . . . . . . . . . . . . . 14 (𝐻 Fn (𝑠 × 𝑠) → dom 𝐻 = (𝑠 × 𝑠))
98adantl 481 . . . . . . . . . . . . 13 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑠 × 𝑠))
109dmeqd 5358 . . . . . . . . . . . 12 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = dom (𝑠 × 𝑠))
11 dmxpid 5377 . . . . . . . . . . . 12 dom (𝑠 × 𝑠) = 𝑠
1210, 11syl6eq 2701 . . . . . . . . . . 11 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = 𝑠)
137, 12eqtr2d 2686 . . . . . . . . . 10 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝑠 = 𝑆)
1413sqxpeqd 5175 . . . . . . . . 9 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → (𝑠 × 𝑠) = (𝑆 × 𝑆))
1514fneq2d 6020 . . . . . . . 8 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝐻 Fn (𝑆 × 𝑆)))
165, 15mpbid 222 . . . . . . 7 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑆 × 𝑆))
1716ex 449 . . . . . 6 (𝜑 → (𝐻 Fn (𝑠 × 𝑠) → 𝐻 Fn (𝑆 × 𝑆)))
184, 17syl5 34 . . . . 5 (𝜑 → (𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 Fn (𝑆 × 𝑆)))
1918rexlimdvw 3063 . . . 4 (𝜑 → (∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 Fn (𝑆 × 𝑆)))
2019adantld 482 . . 3 (𝜑 → ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → 𝐻 Fn (𝑆 × 𝑆)))
2120exlimdv 1901 . 2 (𝜑 → (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → 𝐻 Fn (𝑆 × 𝑆)))
223, 21mpd 15 1 (𝜑𝐻 Fn (𝑆 × 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  𝒫 cpw 4191   class class class wbr 4685   × cxp 5141  dom cdm 5143   Fn wfn 5921  cfv 5926  Xcixp 7950  cat cssc 16514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ixp 7951  df-ssc 16517
This theorem is referenced by:  ssctr  16532  ssceq  16533  subcfn  16548  subsubc  16560
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