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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/c 𝑅 = ∘f/c 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq 7426 | . . . 4 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑥)𝑅𝑐) = ((𝑓‘𝑥)𝑆𝑐)) | |
2 | 1 | mpteq2dv 5250 | . . 3 ⊢ (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐))) |
3 | 2 | mpoeq3dv 7499 | . 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 33715 | . 2 ⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐))) | |
5 | df-ofc 33715 | . 2 ⊢ ∘f/c 𝑆 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐))) | |
6 | 3, 4, 5 | 3eqtr4g 2793 | 1 ⊢ (𝑅 = 𝑆 → ∘f/c 𝑅 = ∘f/c 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 Vcvv 3471 ↦ cmpt 5231 dom cdm 5678 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 ∘f/c cofc 33714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1087 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 df-in 3954 df-ss 3964 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-iota 6500 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-ofc 33715 |
This theorem is referenced by: (None) |
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