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Mirrors > Home > MPE Home > Th. List > caofref | Structured version Visualization version 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 6359 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) |
3 | caofref.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥𝑅𝑥) | |
4 | 3 | ralrimiva 2966 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝑥𝑅𝑥) |
5 | 4 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 𝑥𝑅𝑥) |
6 | id 22 | . . . . . 6 ⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤)) | |
7 | 6, 6 | breq12d 4666 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑥 ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑤))) |
8 | 7 | rspcv 3305 | . . . 4 ⊢ ((𝐹‘𝑤) ∈ 𝑆 → (∀𝑥 ∈ 𝑆 𝑥𝑅𝑥 → (𝐹‘𝑤)𝑅(𝐹‘𝑤))) |
9 | 2, 5, 8 | sylc 65 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤)𝑅(𝐹‘𝑤)) |
10 | 9 | ralrimiva 2966 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐹‘𝑤)) |
11 | ffn 6045 | . . . 4 ⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴) | |
12 | 1, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
13 | caofref.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
14 | inidm 3822 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
15 | eqidd 2623 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤)) | |
16 | 12, 12, 13, 13, 14, 15, 15 | ofrfval 6905 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐹 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐹‘𝑤))) |
17 | 10, 16 | mpbird 247 | 1 ⊢ (𝜑 → 𝐹 ∘𝑟 𝑅𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 class class class wbr 4653 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 ∘𝑟 cofr 6896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ofr 6898 |
This theorem is referenced by: psrridm 19404 itg2itg1 23503 itg20 23504 |
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