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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 17086 | . . 3 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥))) | |
3 | 1, 2 | sylib 220 | . 2 ⊢ (𝜑 → ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥))) |
4 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑡 × 𝑡)) | |
5 | sscfn2.2 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑇 = dom dom 𝐽) | |
6 | 5 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝑇 = dom dom 𝐽) |
7 | fndm 6457 | . . . . . . . . . . . 12 ⊢ (𝐽 Fn (𝑡 × 𝑡) → dom 𝐽 = (𝑡 × 𝑡)) | |
8 | 7 | adantl 484 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom 𝐽 = (𝑡 × 𝑡)) |
9 | 8 | dmeqd 5776 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = dom (𝑡 × 𝑡)) |
10 | dmxpid 5802 | . . . . . . . . . 10 ⊢ dom (𝑡 × 𝑡) = 𝑡 | |
11 | 9, 10 | syl6eq 2874 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = 𝑡) |
12 | 6, 11 | eqtr2d 2859 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝑡 = 𝑇) |
13 | 12 | sqxpeqd 5589 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇)) |
14 | 13 | fneq2d 6449 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑇 × 𝑇))) |
15 | 4, 14 | mpbid 234 | . . . . 5 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑇 × 𝑇)) |
16 | 15 | ex 415 | . . . 4 ⊢ (𝜑 → (𝐽 Fn (𝑡 × 𝑡) → 𝐽 Fn (𝑇 × 𝑇))) |
17 | 16 | adantrd 494 | . . 3 ⊢ (𝜑 → ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥)) → 𝐽 Fn (𝑇 × 𝑇))) |
18 | 17 | exlimdv 1934 | . 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 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∃wrex 3141 𝒫 cpw 4541 class class class wbr 5068 × cxp 5555 dom cdm 5557 Fn wfn 6352 ‘cfv 6357 Xcixp 8463 ⊆cat cssc 17079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ixp 8464 df-ssc 17082 |
This theorem is referenced by: ssc2 17094 ssctr 17097 |
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