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| Mirrors > Home > MPE Home > Th. List > sscfn1 | 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) | 
| sscfn1.2 | ⊢ (𝜑 → 𝑆 = dom dom 𝐻) | 
| Ref | Expression | 
|---|---|
| sscfn1 | ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sscfn1.1 | . . 3 ⊢ (𝜑 → 𝐻 ⊆cat 𝐽) | |
| 2 | brssc 17858 | . . 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 8943 | . . . . . 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 6671 | . . . . . . . . . . . . . 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 5916 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = dom (𝑠 × 𝑠)) | 
| 11 | dmxpid 5941 | . . . . . . . . . . . 12 ⊢ dom (𝑠 × 𝑠) = 𝑠 | |
| 12 | 10, 11 | eqtrdi 2793 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → dom dom 𝐻 = 𝑠) | 
| 13 | 7, 12 | eqtr2d 2778 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → 𝑠 = 𝑆) | 
| 14 | 13 | sqxpeqd 5717 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐻 Fn (𝑠 × 𝑠)) → (𝑠 × 𝑠) = (𝑆 × 𝑆)) | 
| 15 | 14 | fneq2d 6662 | . . . . . . . 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 3160 | . . . 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)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 𝒫 cpw 4600 class class class wbr 5143 × cxp 5683 dom cdm 5685 Fn wfn 6556 ‘cfv 6561 Xcixp 8937 ⊆cat cssc 17851 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ixp 8938 df-ssc 17854 | 
| This theorem is referenced by: ssctr 17869 ssceq 17870 subcfn 17886 subsubc 17898 | 
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