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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 5041 | . . . 4 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ (𝑓‘𝑥)𝑆(𝑔‘𝑥))) | |
2 | 1 | ralbidv 3108 | . . 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 5105 | . 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 7448 | . 2 ⊢ ∘r 𝑅 = {〈𝑓, 𝑔〉 ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)} | |
5 | df-ofr 7448 | . 2 ⊢ ∘r 𝑆 = {〈𝑓, 𝑔〉 ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑆(𝑔‘𝑥)} | |
6 | 3, 4, 5 | 3eqtr4g 2796 | 1 ⊢ (𝑅 = 𝑆 → ∘r 𝑅 = ∘r 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∀wral 3051 ∩ cin 3852 class class class wbr 5039 {copab 5101 dom cdm 5536 ‘cfv 6358 ∘r cofr 7446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-br 5040 df-opab 5102 df-ofr 7448 |
This theorem is referenced by: (None) |
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