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Mirrors > Home > MPE Home > Th. List > ofeqd | Structured version Visualization version GIF version |
Description: Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.) |
Ref | Expression |
---|---|
ofeqd.1 | ⊢ (𝜑 → 𝑅 = 𝑆) |
Ref | Expression |
---|---|
ofeqd | ⊢ (𝜑 → ∘f 𝑅 = ∘f 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ofeqd.1 | . . . . 5 ⊢ (𝜑 → 𝑅 = 𝑆) | |
2 | 1 | oveqd 7289 | . . . 4 ⊢ (𝜑 → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝑓‘𝑥)𝑆(𝑔‘𝑥))) |
3 | 2 | mpteq2dv 5181 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))) |
4 | 3 | mpoeq3dv 7349 | . 2 ⊢ (𝜑 → (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥))))) |
5 | df-of 7528 | . 2 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | |
6 | df-of 7528 | . 2 ⊢ ∘f 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))) | |
7 | 4, 5, 6 | 3eqtr4g 2805 | 1 ⊢ (𝜑 → ∘f 𝑅 = ∘f 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Vcvv 3431 ∩ cin 3891 ↦ cmpt 5162 dom cdm 5590 ‘cfv 6432 (class class class)co 7272 ∈ cmpo 7274 ∘f cof 7526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1088 df-tru 1545 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-v 3433 df-in 3899 df-ss 3909 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-iota 6390 df-fv 6440 df-ov 7275 df-oprab 7276 df-mpo 7277 df-of 7528 |
This theorem is referenced by: ofeq 7531 psrval 21129 resspsradd 21196 fedgmullem1 31719 fedgmullem2 31720 sitmval 32325 ldualset 37148 mendval 41017 |
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