Theorem sscfn2 17780

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 17776	. . 3 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥)))
3	1, 2	sylib 218	. 2 ⊢ (𝜑 → ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥)))
4		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑡 × 𝑡))
5		sscfn2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 = dom dom 𝐽)
6	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝑇 = dom dom 𝐽)
7		fndm 6621	. . . . . . . . . . . 12 ⊢ (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡))
8	7	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑡 × 𝑡))
9	8	dmeqd 5869	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = dom (𝑡 × 𝑡))
10		dmxpid 5894	. . . . . . . . . 10 ⊢ dom (𝑡 × 𝑡) = 𝑡
11	9, 10	eqtrdi 2780	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = 𝑡)
12	6, 11	eqtr2d 2765	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝑡 = 𝑇)
13	12	sqxpeqd 5670	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇))
14	13	fneq2d 6612	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑇 × 𝑇)))
15	4, 14	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑇 × 𝑇))
16	15	ex 412	. . . 4 ⊢ (𝜑 → (𝐽 Fn (𝑡 × 𝑡) → 𝐽 Fn (𝑇 × 𝑇)))
17	16	adantrd 491	. . 3 ⊢ (𝜑 → ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥)) → 𝐽 Fn (𝑇 × 𝑇)))
18	17	exlimdv 1933	. 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 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053  𝒫 cpw 4563   class class class wbr 5107   × cxp 5636  dom cdm 5638   Fn wfn 6506  ‘cfv 6511  Xcixp 8870   ⊆_cat cssc 17769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ixp 8871  df-ssc 17772

This theorem is referenced by:  ssc2  17784  ssctr  17787  iinfssc  49046