Theorem sscfn1 17753

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 17750	. . 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 5862	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = dom (𝑠 × 𝑠))
11		dmxpid 5887	. . . . . . . . . . . 12 ⊢ dom (𝑠 × 𝑠) = 𝑠
12	10, 11	eqtrdi 2788	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = 𝑠)
13	7, 12	eqtr2d 2773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝑠 = 𝑆)
14	13	sqxpeqd 5664	. . . . . . . . 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 3144	. . . 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 1935	. 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 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3062  𝒫 cpw 4556   class class class wbr 5100   × cxp 5630  dom cdm 5632   Fn wfn 6495  ‘cfv 6500  Xcixp 8847   ⊆_cat cssc 17743

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ixp 8848  df-ssc 17746

This theorem is referenced by:  ssctr  17761  ssceq  17762  subcfn  17777  subsubc  17789  iinfssclem1  49413  iinfssc  49416