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Theorem sscfn1 17147
 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 17144 . . 3 (𝐻cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
31, 2sylib 221 . 2 (𝜑 → ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)))
4 ixpfn 8486 . . . . . 6 (𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 Fn (𝑠 × 𝑠))
5 simpr 489 . . . . . . . 8 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑠 × 𝑠))
6 sscfn1.2 . . . . . . . . . . . 12 (𝜑𝑆 = dom dom 𝐻)
76adantr 485 . . . . . . . . . . 11 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝑆 = dom dom 𝐻)
8 fndm 6437 . . . . . . . . . . . . . 14 (𝐻 Fn (𝑠 × 𝑠) → dom 𝐻 = (𝑠 × 𝑠))
98adantl 486 . . . . . . . . . . . . 13 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑠 × 𝑠))
109dmeqd 5746 . . . . . . . . . . . 12 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = dom (𝑠 × 𝑠))
11 dmxpid 5772 . . . . . . . . . . . 12 dom (𝑠 × 𝑠) = 𝑠
1210, 11eqtrdi 2810 . . . . . . . . . . 11 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = 𝑠)
137, 12eqtr2d 2795 . . . . . . . . . 10 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝑠 = 𝑆)
1413sqxpeqd 5557 . . . . . . . . 9 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → (𝑠 × 𝑠) = (𝑆 × 𝑆))
1514fneq2d 6429 . . . . . . . 8 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝐻 Fn (𝑆 × 𝑆)))
165, 15mpbid 235 . . . . . . 7 ((𝜑𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑆 × 𝑆))
1716ex 417 . . . . . 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 3215 . . . 4 (𝜑 → (∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥) → 𝐻 Fn (𝑆 × 𝑆)))
2019adantld 495 . . 3 (𝜑 → ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽𝑥)) → 𝐻 Fn (𝑆 × 𝑆)))
2120exlimdv 1935 . 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 400   = wceq 1539  ∃wex 1782   ∈ wcel 2112  ∃wrex 3072  𝒫 cpw 4495   class class class wbr 5033   × cxp 5523  dom cdm 5525   Fn wfn 6331  ‘cfv 6336  Xcixp 8480   ⊆cat cssc 17137 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ixp 8481  df-ssc 17140 This theorem is referenced by:  ssctr  17155  ssceq  17156  subcfn  17171  subsubc  17183
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