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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 7425 | . . . 4 ⊢ (𝜑 → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝑓‘𝑥)𝑆(𝑔‘𝑥))) |
3 | 2 | mpteq2dv 5250 | . . 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 7487 | . 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 7669 | . 2 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | |
6 | df-of 7669 | . 2 ⊢ ∘f 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))) | |
7 | 4, 5, 6 | 3eqtr4g 2797 | 1 ⊢ (𝜑 → ∘f 𝑅 = ∘f 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 Vcvv 3474 ∩ cin 3947 ↦ cmpt 5231 dom cdm 5676 ‘cfv 6543 (class class class)co 7408 ∈ cmpo 7410 ∘f cof 7667 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-in 3955 df-ss 3965 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-iota 6495 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 |
This theorem is referenced by: ofeq 7672 psrval 21467 resspsradd 21535 fedgmullem1 32709 fedgmullem2 32710 sitmval 33343 ldualset 37990 mendval 41915 |
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