![]() |
Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > colleq12d | Structured version Visualization version GIF version |
Description: Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.) |
Ref | Expression |
---|---|
colleq12d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
colleq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
colleq12d | ⊢ (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | colleq12d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | colleq12d.1 | . . . . 5 ⊢ (𝜑 → 𝐹 = 𝐺) | |
3 | 2 | imaeq1d 6079 | . . . 4 ⊢ (𝜑 → (𝐹 “ {𝑥}) = (𝐺 “ {𝑥})) |
4 | 3 | scotteqd 44233 | . . 3 ⊢ (𝜑 → Scott (𝐹 “ {𝑥}) = Scott (𝐺 “ {𝑥})) |
5 | 1, 4 | iuneq12d 5026 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥})) |
6 | df-coll 44247 | . 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) | |
7 | df-coll 44247 | . 2 ⊢ (𝐺 Coll 𝐵) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥}) | |
8 | 5, 6, 7 | 3eqtr4g 2800 | 1 ⊢ (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 {csn 4631 ∪ ciun 4996 “ cima 5692 Scott cscott 44231 Coll ccoll 44246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-iun 4998 df-br 5149 df-opab 5211 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-scott 44232 df-coll 44247 |
This theorem is referenced by: colleq1 44250 colleq2 44251 |
Copyright terms: Public domain | W3C validator |