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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 7448 | . . . 4 ⊢ (𝜑 → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝑓‘𝑥)𝑆(𝑔‘𝑥))) |
| 3 | 2 | mpteq2dv 5244 | . . 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 7512 | . 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 7697 | . 2 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | |
| 6 | df-of 7697 | . 2 ⊢ ∘f 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))) | |
| 7 | 4, 5, 6 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → ∘f 𝑅 = ∘f 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 Vcvv 3480 ∩ cin 3950 ↦ cmpt 5225 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ∘f cof 7695 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-iota 6514 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 |
| This theorem is referenced by: ofeq 7700 psrval 21935 resspsradd 21995 elrgspnlem1 33246 fedgmullem1 33680 fedgmullem2 33681 sitmval 34351 ldualset 39126 mendval 43191 |
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