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| Mirrors > Home > MPE Home > Th. List > ofreq | Structured version Visualization version GIF version | ||
| Description: Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
| Ref | Expression |
|---|---|
| ofreq | ⊢ (𝑅 = 𝑆 → ∘r 𝑅 = ∘r 𝑆) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | breq 5145 | . . . 4 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ (𝑓‘𝑥)𝑆(𝑔‘𝑥))) | |
| 2 | 1 | ralbidv 3178 | . . 3 ⊢ (𝑅 = 𝑆 → (∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑆(𝑔‘𝑥))) |
| 3 | 2 | opabbidv 5209 | . 2 ⊢ (𝑅 = 𝑆 → {〈𝑓, 𝑔〉 ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)} = {〈𝑓, 𝑔〉 ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑆(𝑔‘𝑥)}) |
| 4 | df-ofr 7698 | . 2 ⊢ ∘r 𝑅 = {〈𝑓, 𝑔〉 ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)} | |
| 5 | df-ofr 7698 | . 2 ⊢ ∘r 𝑆 = {〈𝑓, 𝑔〉 ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑆(𝑔‘𝑥)} | |
| 6 | 3, 4, 5 | 3eqtr4g 2802 | 1 ⊢ (𝑅 = 𝑆 → ∘r 𝑅 = ∘r 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∀wral 3061 ∩ cin 3950 class class class wbr 5143 {copab 5205 dom cdm 5685 ‘cfv 6561 ∘r cofr 7696 |
| 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-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-br 5144 df-opab 5206 df-ofr 7698 |
| This theorem is referenced by: (None) |
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