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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 1132 | . . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝑓 ∈ V ∧ 𝑔 ∈ V) → 𝑅 = 𝑆) | |
2 | 1 | oveqd 7167 | . . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝑓 ∈ V ∧ 𝑔 ∈ V) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝑓‘𝑥)𝑆(𝑔‘𝑥))) |
3 | 2 | mpteq2dv 5155 | . . 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 7225 | . 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 7403 | . 2 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | |
6 | df-of 7403 | . 2 ⊢ ∘f 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))) | |
7 | 4, 5, 6 | 3eqtr4g 2881 | 1 ⊢ (𝑅 = 𝑆 → ∘f 𝑅 = ∘f 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∩ cin 3935 ↦ cmpt 5139 dom cdm 5550 ‘cfv 6350 (class class class)co 7150 ∈ cmpo 7152 ∘f cof 7401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-ral 3143 df-rex 3144 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-iota 6309 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 |
This theorem is referenced by: psrval 20136 resspsradd 20190 resspsrvsca 20192 fedgmullem1 31020 fedgmullem2 31021 sitmval 31602 ldualset 36255 mendval 39776 mendplusgfval 39778 mendvscafval 39783 |
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