Theorem sscfn1 17755

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)
sscfn1.2	⊢ (𝜑 → 𝑆 = dom dom 𝐻)

Assertion

Ref	Expression
sscfn1	⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))

Proof of Theorem sscfn1

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

Step	Hyp	Ref	Expression
1		sscfn1.1	. . 3 ⊢ (𝜑 → 𝐻 ⊆cat 𝐽)
2		brssc 17752	. . 3 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))
3	1, 2	sylib 218	. 2 ⊢ (𝜑 → ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)))
4		ixpfn 8853	. . . . . 6 ⊢ (𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) → 𝐻 Fn (𝑠 × 𝑠))
5		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑠 × 𝑠))
6		sscfn1.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 = dom dom 𝐻)
7	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝑆 = dom dom 𝐻)
8		fndm 6603	. . . . . . . . . . . . . 14 ⊢ (𝐻 Fn (𝑠 × 𝑠) → dom 𝐻 = (𝑠 × 𝑠))
9	8	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom 𝐻 = (𝑠 × 𝑠))
10	9	dmeqd 5859	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = dom (𝑠 × 𝑠))
11		dmxpid 5883	. . . . . . . . . . . 12 ⊢ dom (𝑠 × 𝑠) = 𝑠
12	10, 11	eqtrdi 2780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = 𝑠)
13	7, 12	eqtr2d 2765	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝑠 = 𝑆)
14	13	sqxpeqd 5663	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → (𝑠 × 𝑠) = (𝑆 × 𝑆))
15	14	fneq2d 6594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → (𝐻 Fn (𝑠 × 𝑠) ↔ 𝐻 Fn (𝑆 × 𝑆)))
16	5, 15	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝐻 Fn (𝑆 × 𝑆))
17	16	ex 412	. . . . . 6 ⊢ (𝜑 → (𝐻 Fn (𝑠 × 𝑠) → 𝐻 Fn (𝑆 × 𝑆)))
18	4, 17	syl5 34	. . . . 5 ⊢ (𝜑 → (𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) → 𝐻 Fn (𝑆 × 𝑆)))
19	18	rexlimdvw 3139	. . . 4 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥) → 𝐻 Fn (𝑆 × 𝑆)))
20	19	adantld 490	. . 3 ⊢ (𝜑 → ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)) → 𝐻 Fn (𝑆 × 𝑆)))
21	20	exlimdv 1933	. 2 ⊢ (𝜑 → (∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑠 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑠 × 𝑠)𝒫 (𝐽‘𝑥)) → 𝐻 Fn (𝑆 × 𝑆)))
22	3, 21	mpd 15	1 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053  𝒫 cpw 4559   class class class wbr 5102   × cxp 5629  dom cdm 5631   Fn wfn 6494  ‘cfv 6499  Xcixp 8847   ⊆_cat cssc 17745

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ixp 8848  df-ssc 17748

This theorem is referenced by:  ssctr  17763  ssceq  17764  subcfn  17779  subsubc  17791  iinfssclem1  49016  iinfssc  49019