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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofceq | Structured version Visualization version GIF version |
Description: Equality theorem for function/constant operation. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
Ref | Expression |
---|---|
ofceq | ⊢ (𝑅 = 𝑆 → ∘𝑓/𝑐𝑅 = ∘𝑓/𝑐𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq 6980 | . . . 4 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑥)𝑅𝑐) = ((𝑓‘𝑥)𝑆𝑐)) | |
2 | 1 | mpteq2dv 5019 | . . 3 ⊢ (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐))) |
3 | 2 | mpoeq3dv 7049 | . 2 ⊢ (𝑅 = 𝑆 → (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐))) = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐)))) |
4 | df-ofc 31031 | . 2 ⊢ ∘𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐))) | |
5 | df-ofc 31031 | . 2 ⊢ ∘𝑓/𝑐𝑆 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐))) | |
6 | 3, 4, 5 | 3eqtr4g 2832 | 1 ⊢ (𝑅 = 𝑆 → ∘𝑓/𝑐𝑅 = ∘𝑓/𝑐𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 Vcvv 3408 ↦ cmpt 5004 dom cdm 5403 ‘cfv 6185 (class class class)co 6974 ∈ cmpo 6976 ∘𝑓/𝑐cofc 31030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-12 2107 ax-ext 2743 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-ral 3086 df-rex 3087 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-iota 6149 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-ofc 31031 |
This theorem is referenced by: (None) |
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