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Mirrors > Home > MPE Home > Th. List > ofeq | Structured version Visualization version GIF version |
Description: Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.) |
Ref | Expression |
---|---|
ofeq | ⊢ (𝑅 = 𝑆 → ∘f 𝑅 = ∘f 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1133 | . . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝑓 ∈ V ∧ 𝑔 ∈ V) → 𝑅 = 𝑆) | |
2 | 1 | oveqd 7152 | . . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝑓 ∈ V ∧ 𝑔 ∈ V) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝑓‘𝑥)𝑆(𝑔‘𝑥))) |
3 | 2 | mpteq2dv 5126 | . . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝑓 ∈ V ∧ 𝑔 ∈ V) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))) |
4 | 3 | mpoeq3dva 7210 | . 2 ⊢ (𝑅 = 𝑆 → (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 7389 | . 2 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | |
6 | df-of 7389 | . 2 ⊢ ∘f 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))) | |
7 | 4, 5, 6 | 3eqtr4g 2858 | 1 ⊢ (𝑅 = 𝑆 → ∘f 𝑅 = ∘f 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 ↦ cmpt 5110 dom cdm 5519 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ∘f cof 7387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-in 3888 df-ss 3898 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-iota 6283 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 |
This theorem is referenced by: psrval 20600 resspsradd 20654 resspsrvsca 20656 fedgmullem1 31113 fedgmullem2 31114 sitmval 31717 ldualset 36421 mendval 40127 mendplusgfval 40129 mendvscafval 40134 |
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