Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > colleq2 | Structured version Visualization version GIF version |
Description: Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
colleq2 | ⊢ (𝐴 = 𝐵 → (𝐹 Coll 𝐴) = (𝐹 Coll 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2822 | . 2 ⊢ (𝐴 = 𝐵 → 𝐹 = 𝐹) | |
2 | id 22 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) | |
3 | 1, 2 | colleq12d 40679 | 1 ⊢ (𝐴 = 𝐵 → (𝐹 Coll 𝐴) = (𝐹 Coll 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Coll ccoll 40676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-iun 4907 df-br 5053 df-opab 5115 df-cnv 5549 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-scott 40662 df-coll 40677 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |