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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 6656 | . . . 4 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑥)𝑅𝑐) = ((𝑓‘𝑥)𝑆𝑐)) | |
2 | 1 | mpteq2dv 4745 | . . 3 ⊢ (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐))) |
3 | 2 | mpt2eq3dv 6721 | . 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 30158 | . 2 ⊢ ∘𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐))) | |
5 | df-ofc 30158 | . 2 ⊢ ∘𝑓/𝑐𝑆 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐))) | |
6 | 3, 4, 5 | 3eqtr4g 2681 | 1 ⊢ (𝑅 = 𝑆 → ∘𝑓/𝑐𝑅 = ∘𝑓/𝑐𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 Vcvv 3200 ↦ cmpt 4729 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ∘𝑓/𝑐cofc 30157 |
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 |
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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-iota 5851 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-ofc 30158 |
This theorem is referenced by: (None) |
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