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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/c 𝑅 = ∘f/c 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq 7156 | . . . 4 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑥)𝑅𝑐) = ((𝑓‘𝑥)𝑆𝑐)) | |
2 | 1 | mpteq2dv 5154 | . . 3 ⊢ (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐))) |
3 | 2 | mpoeq3dv 7227 | . 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 31350 | . 2 ⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐))) | |
5 | df-ofc 31350 | . 2 ⊢ ∘f/c 𝑆 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐))) | |
6 | 3, 4, 5 | 3eqtr4g 2881 | 1 ⊢ (𝑅 = 𝑆 → ∘f/c 𝑅 = ∘f/c 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Vcvv 3494 ↦ cmpt 5138 dom cdm 5549 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 ∘f/c cofc 31349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-ral 3143 df-rex 3144 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-iota 6308 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-ofc 31350 |
This theorem is referenced by: (None) |
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