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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 5323 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) |
3 | caofref.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥𝑅𝑥) | |
4 | 3 | ralrimiva 2434 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝑥𝑅𝑥) |
5 | 4 | adantr 270 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 𝑥𝑅𝑥) |
6 | id 19 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤)) | |
7 | 6, 6 | breq12d 3798 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑥 ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑤))) |
8 | 7 | rspcv 2697 | . . . 4 ⊢ ((𝐹‘𝑤) ∈ 𝑆 → (∀𝑥 ∈ 𝑆 𝑥𝑅𝑥 → (𝐹‘𝑤)𝑅(𝐹‘𝑤))) |
9 | 2, 5, 8 | sylc 61 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤)𝑅(𝐹‘𝑤)) |
10 | 9 | ralrimiva 2434 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐹‘𝑤)) |
11 | ffn 5066 | . . . 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 3175 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
15 | eqidd 2082 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤)) | |
16 | 12, 12, 13, 13, 14, 15, 15 | ofrfval 5740 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐹 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐹‘𝑤))) |
17 | 10, 16 | mpbird 165 | 1 ⊢ (𝜑 → 𝐹 ∘𝑟 𝑅𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 ∀wral 2348 class class class wbr 3785 Fn wfn 4917 ⟶wf 4918 ‘cfv 4922 ∘𝑟 cofr 5731 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ofr 5733 |
This theorem is referenced by: (None) |
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