Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5071 | . . . 4 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ (𝑓‘𝑥)𝑆(𝑔‘𝑥))) | |
2 | 1 | ralbidv 3200 | . . 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 5135 | . 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 7413 | . 2 ⊢ ∘r 𝑅 = {〈𝑓, 𝑔〉 ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)} | |
5 | df-ofr 7413 | . 2 ⊢ ∘r 𝑆 = {〈𝑓, 𝑔〉 ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑆(𝑔‘𝑥)} | |
6 | 3, 4, 5 | 3eqtr4g 2884 | 1 ⊢ (𝑅 = 𝑆 → ∘r 𝑅 = ∘r 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∀wral 3141 ∩ cin 3938 class class class wbr 5069 {copab 5131 dom cdm 5558 ‘cfv 6358 ∘r cofr 7411 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-ral 3146 df-br 5070 df-opab 5132 df-ofr 7413 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |