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Mirrors > Home > ILE Home > Th. List > caofref | GIF version |
Description: Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
caofref.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
caofref.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
caofref.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥𝑅𝑥) |
Ref | Expression |
---|---|
caofref | ⊢ (𝜑 → 𝐹 ∘𝑟 𝑅𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caofref.2 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
2 | 1 | ffvelrnda 5302 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) |
3 | caofref.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥𝑅𝑥) | |
4 | 3 | ralrimiva 2392 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝑥𝑅𝑥) |
5 | 4 | adantr 261 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 𝑥𝑅𝑥) |
6 | id 19 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤)) | |
7 | 6, 6 | breq12d 3777 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑥 ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑤))) |
8 | 7 | rspcv 2652 | . . . 4 ⊢ ((𝐹‘𝑤) ∈ 𝑆 → (∀𝑥 ∈ 𝑆 𝑥𝑅𝑥 → (𝐹‘𝑤)𝑅(𝐹‘𝑤))) |
9 | 2, 5, 8 | sylc 56 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤)𝑅(𝐹‘𝑤)) |
10 | 9 | ralrimiva 2392 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐹‘𝑤)) |
11 | ffn 5046 | . . . 4 ⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴) | |
12 | 1, 11 | syl 14 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
13 | caofref.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
14 | inidm 3146 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
15 | eqidd 2041 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤)) | |
16 | 12, 12, 13, 13, 14, 15, 15 | ofrfval 5720 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐹 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐹‘𝑤))) |
17 | 10, 16 | mpbird 156 | 1 ⊢ (𝜑 → 𝐹 ∘𝑟 𝑅𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 ∀wral 2306 class class class wbr 3764 Fn wfn 4897 ⟶wf 4898 ‘cfv 4902 ∘𝑟 cofr 5711 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ofr 5713 |
This theorem is referenced by: (None) |
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