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Theorem sscfn1 16300
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 16297 . . 3 (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
31, 2sylib 207 . 2 (𝜑 → ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
4 ixpfn 7800 . . . . . 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 5904 . . . . . . . . . . . . . 14 (𝐻 Fn (𝑠 × 𝑠) → dom 𝐻 = (𝑠 × 𝑠))
98adantl 481 . . . . . . . . . . . . 13 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑠 × 𝑠))
109dmeqd 5248 . . . . . . . . . . . 12 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = dom (𝑠 × 𝑠))
11 dmxpid 5266 . . . . . . . . . . . 12 dom (𝑠 × 𝑠) = 𝑠
1210, 11syl6eq 2660 . . . . . . . . . . 11 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = 𝑠)
137, 12eqtr2d 2645 . . . . . . . . . 10 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝑠 = 𝑆)
1413sqxpeqd 5065 . . . . . . . . 9 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → (𝑠 × 𝑠) = (𝑆 × 𝑆))
1514fneq2d 5896 . . . . . . . 8 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝐻 Fn (𝑆 × 𝑆)))
165, 15mpbid 221 . . . . . . 7 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑆 × 𝑆))
1716ex 449 . . . . . 6 (𝜑 → (𝐻 Fn (𝑠 × 𝑠) → 𝐻 Fn (𝑆 × 𝑆)))
184, 17syl5 33 . . . . 5 (𝜑 → (𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 Fn (𝑆 × 𝑆)))
1918rexlimdvw 3016 . . . 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 1848 . 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 1475  wex 1695  wcel 1977  wrex 2897  𝒫 cpw 4108   class class class wbr 4583   × cxp 5036  dom cdm 5038   Fn wfn 5799  cfv 5804  Xcixp 7794  cat cssc 16290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ixp 7795  df-ssc 16293
This theorem is referenced by:  ssctr  16308  ssceq  16309  subcfn  16324  subsubc  16336
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