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Mirrors > Home > MPE Home > Th. List > sscfn2 | Structured version Visualization version GIF version |
Description: The subcategory subset relation is defined on functions with square domain. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
sscfn1.1 | ⊢ (𝜑 → 𝐻 ⊆cat 𝐽) |
sscfn2.2 | ⊢ (𝜑 → 𝑇 = dom dom 𝐽) |
Ref | Expression |
---|---|
sscfn2 | ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sscfn1.1 | . . 3 ⊢ (𝜑 → 𝐻 ⊆cat 𝐽) | |
2 | brssc 16936 | . . 3 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥))) | |
3 | 1, 2 | sylib 210 | . 2 ⊢ (𝜑 → ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥))) |
4 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑡 × 𝑡)) | |
5 | sscfn2.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑇 = dom dom 𝐽) | |
6 | 5 | adantr 473 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝑇 = dom dom 𝐽) |
7 | fndm 6282 | . . . . . . . . . . . 12 ⊢ (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡)) | |
8 | 7 | adantl 474 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑡 × 𝑡)) |
9 | 8 | dmeqd 5618 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = dom (𝑡 × 𝑡)) |
10 | dmxpid 5637 | . . . . . . . . . 10 ⊢ dom (𝑡 × 𝑡) = 𝑡 | |
11 | 9, 10 | syl6eq 2824 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = 𝑡) |
12 | 6, 11 | eqtr2d 2809 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝑡 = 𝑇) |
13 | 12 | sqxpeqd 5433 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇)) |
14 | 13 | fneq2d 6274 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑇 × 𝑇))) |
15 | 4, 14 | mpbid 224 | . . . . 5 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑇 × 𝑇)) |
16 | 15 | ex 405 | . . . 4 ⊢ (𝜑 → (𝐽 Fn (𝑡 × 𝑡) → 𝐽 Fn (𝑇 × 𝑇))) |
17 | 16 | adantrd 484 | . . 3 ⊢ (𝜑 → ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥)) → 𝐽 Fn (𝑇 × 𝑇))) |
18 | 17 | exlimdv 1892 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∃wex 1742 ∈ wcel 2050 ∃wrex 3083 𝒫 cpw 4416 class class class wbr 4923 × cxp 5399 dom cdm 5401 Fn wfn 6177 ‘cfv 6182 Xcixp 8253 ⊆cat cssc 16929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5306 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ixp 8254 df-ssc 16932 |
This theorem is referenced by: ssc2 16944 ssctr 16947 |
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