Theorem sscfn2 17744

Description: The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
sscfn1.1	⊢ (𝜑 → 𝐻 ⊆cat 𝐽)
sscfn2.2	⊢ (𝜑 → 𝑇 = dom dom 𝐽)

Assertion

Ref	Expression
sscfn2	⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))

Proof of Theorem sscfn2

Dummy variables 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sscfn1.1	. . 3 ⊢ (𝜑 → 𝐻 ⊆cat 𝐽)
2		brssc 17740	. . 3 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥)))
3	1, 2	sylib 217	. 2 ⊢ (𝜑 → ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥)))
4		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑡 × 𝑡))
5		sscfn2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 = dom dom 𝐽)
6	5	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝑇 = dom dom 𝐽)
7		fndm 6638	. . . . . . . . . . . 12 ⊢ (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
8	7	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑡 × 𝑡))
9	8	dmeqd 5894	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = dom (𝑡 × 𝑡))
10		dmxpid 5918	. . . . . . . . . 10 ⊢ dom (𝑡 × 𝑡) = 𝑡
11	9, 10	eqtrdi 2787	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = 𝑡)
12	6, 11	eqtr2d 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝑡 = 𝑇)
13	12	sqxpeqd 5698	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇))
14	13	fneq2d 6629	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑇 × 𝑇)))
15	4, 14	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑇 × 𝑇))
16	15	ex 413	. . . 4 ⊢ (𝜑 → (𝐽 Fn (𝑡 × 𝑡) → 𝐽 Fn (𝑇 × 𝑇)))
17	16	adantrd 492	. . 3 ⊢ (𝜑 → ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥)) → 𝐽 Fn (𝑇 × 𝑇)))
18	17	exlimdv 1936	. 2 ⊢ (𝜑 → (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥)) → 𝐽 Fn (𝑇 × 𝑇)))
19	3, 18	mpd 15	1 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∃wrex 3069  𝒫 cpw 4593   class class class wbr 5138   × cxp 5664  dom cdm 5666   Fn wfn 6524  ‘cfv 6529  Xcixp 8871   ⊆_cat cssc 17733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7705

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3430  df-v 3472  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6481  df-fun 6531  df-fn 6532  df-f 6533  df-f1 6534  df-fo 6535  df-f1o 6536  df-fv 6537  df-ixp 8872  df-ssc 17736

This theorem is referenced by:  ssc2  17748  ssctr  17751