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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 | id 22 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑤) → 𝑥 = (𝐹‘𝑤)) | |
2 | 1, 1 | breq12d 5066 | . . . 4 ⊢ (𝑥 = (𝐹‘𝑤) → (𝑥𝑅𝑥 ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑤))) |
3 | caofref.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥𝑅𝑥) | |
4 | 3 | ralrimiva 3105 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 𝑥𝑅𝑥) |
5 | 4 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∀𝑥 ∈ 𝑆 𝑥𝑅𝑥) |
6 | caofref.2 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
7 | 6 | ffvelrnda 6904 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) |
8 | 2, 5, 7 | rspcdva 3539 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤)𝑅(𝐹‘𝑤)) |
9 | 8 | ralrimiva 3105 | . 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐹‘𝑤)) |
10 | 6 | ffnd 6546 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
11 | caofref.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
12 | inidm 4133 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
13 | eqidd 2738 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) = (𝐹‘𝑤)) | |
14 | 10, 10, 11, 11, 12, 13, 13 | ofrfval 7478 | . 2 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐹 ↔ ∀𝑤 ∈ 𝐴 (𝐹‘𝑤)𝑅(𝐹‘𝑤))) |
15 | 9, 14 | mpbird 260 | 1 ⊢ (𝜑 → 𝐹 ∘r 𝑅𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 class class class wbr 5053 ⟶wf 6376 ‘cfv 6380 ∘r cofr 7468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ofr 7470 |
This theorem is referenced by: psrridm 20929 itg2itg1 24634 itg20 24635 |
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