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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 5141 | . . . 4 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ (𝑓‘𝑥)𝑆(𝑔‘𝑥))) | |
2 | 1 | ralbidv 3169 | . . 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 5205 | . 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 7665 | . 2 ⊢ ∘r 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)} | |
5 | df-ofr 7665 | . 2 ⊢ ∘r 𝑆 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑆(𝑔‘𝑥)} | |
6 | 3, 4, 5 | 3eqtr4g 2789 | 1 ⊢ (𝑅 = 𝑆 → ∘r 𝑅 = ∘r 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∀wral 3053 ∩ cin 3940 class class class wbr 5139 {copab 5201 dom cdm 5667 ‘cfv 6534 ∘r cofr 7663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-br 5140 df-opab 5202 df-ofr 7665 |
This theorem is referenced by: (None) |
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