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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 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 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 2800 | 1 ⊢ (𝜑 → ∘f 𝑅 = ∘f 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Vcvv 3478 ∩ cin 3962 ↦ cmpt 5231 dom cdm 5689 ‘cfv 6563 (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 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-iota 6516 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 |
This theorem is referenced by: ofeq 7700 psrval 21953 resspsradd 22013 elrgspnlem1 33232 fedgmullem1 33657 fedgmullem2 33658 sitmval 34331 ldualset 39107 mendval 43168 |
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