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Theorem sscfn1 16698
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 16695 . . 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 8082 . . . . . 6 (𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 Fn (𝑠 × 𝑠))
5 simpr 479 . . . . . . . 8 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑠 × 𝑠))
6 sscfn1.2 . . . . . . . . . . . 12 (𝜑𝑆 = dom dom 𝐻)
76adantr 472 . . . . . . . . . . 11 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝑆 = dom dom 𝐻)
8 fndm 6151 . . . . . . . . . . . . . 14 (𝐻 Fn (𝑠 × 𝑠) → dom 𝐻 = (𝑠 × 𝑠))
98adantl 473 . . . . . . . . . . . . 13 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑠 × 𝑠))
109dmeqd 5481 . . . . . . . . . . . 12 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = dom (𝑠 × 𝑠))
11 dmxpid 5500 . . . . . . . . . . . 12 dom (𝑠 × 𝑠) = 𝑠
1210, 11syl6eq 2810 . . . . . . . . . . 11 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = 𝑠)
137, 12eqtr2d 2795 . . . . . . . . . 10 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝑠 = 𝑆)
1413sqxpeqd 5298 . . . . . . . . 9 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → (𝑠 × 𝑠) = (𝑆 × 𝑆))
1514fneq2d 6143 . . . . . . . 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 3172 . . . 4 (𝜑 → (∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 Fn (𝑆 × 𝑆)))
2019adantld 484 . . 3 (𝜑 → ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → 𝐻 Fn (𝑆 × 𝑆)))
2120exlimdv 2010 . 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 1632  wex 1853  wcel 2139  wrex 3051  𝒫 cpw 4302   class class class wbr 4804   × cxp 5264  dom cdm 5266   Fn wfn 6044  cfv 6049  Xcixp 8076  cat cssc 16688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ixp 8077  df-ssc 16691
This theorem is referenced by:  ssctr  16706  ssceq  16707  subcfn  16722  subsubc  16734
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