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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 16297 | . . 3 ⊢ (𝐻 ⊆cat 𝐽 ↔ ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥))) | |
3 | 1, 2 | sylib 207 | . 2 ⊢ (𝜑 → ∃𝑡(𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥))) |
4 | simpr 476 | . . . . . 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 5904 | . . . . . . . . . . . 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 5248 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = dom (𝑡 × 𝑡)) |
10 | dmxpid 5266 | . . . . . . . . . 10 ⊢ dom (𝑡 × 𝑡) = 𝑡 | |
11 | 9, 10 | syl6eq 2660 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → dom dom 𝐽 = 𝑡) |
12 | 6, 11 | eqtr2d 2645 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝑡 = 𝑇) |
13 | 12 | sqxpeqd 5065 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝑡 × 𝑡) = (𝑇 × 𝑇)) |
14 | 13 | fneq2d 5896 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → (𝐽 Fn (𝑡 × 𝑡) ↔ 𝐽 Fn (𝑇 × 𝑇))) |
15 | 4, 14 | mpbid 221 | . . . . 5 ⊢ ((𝜑 ∧ 𝐽 Fn (𝑡 × 𝑡)) → 𝐽 Fn (𝑇 × 𝑇)) |
16 | 15 | ex 449 | . . . 4 ⊢ (𝜑 → (𝐽 Fn (𝑡 × 𝑡) → 𝐽 Fn (𝑇 × 𝑇))) |
17 | 16 | adantrd 483 | . . 3 ⊢ (𝜑 → ((𝐽 Fn (𝑡 × 𝑡) ∧ ∃𝑦 ∈ 𝒫 𝑡𝐻 ∈ X𝑥 ∈ (𝑦 × 𝑦)𝒫 (𝐽‘𝑥)) → 𝐽 Fn (𝑇 × 𝑇))) |
18 | 17 | exlimdv 1848 | . 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 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ∃wrex 2897 𝒫 cpw 4108 class class class wbr 4583 × cxp 5036 dom cdm 5038 Fn wfn 5799 ‘cfv 5804 Xcixp 7794 ⊆cat cssc 16290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ixp 7795 df-ssc 16293 |
This theorem is referenced by: ssc2 16305 ssctr 16308 |
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