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| 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 6076 | . . . 4 ⊢ (𝜑 → (𝐹 “ {𝑥}) = (𝐺 “ {𝑥})) | 
| 4 | 3 | scotteqd 44261 | . . 3 ⊢ (𝜑 → Scott (𝐹 “ {𝑥}) = Scott (𝐺 “ {𝑥})) | 
| 5 | 1, 4 | iuneq12d 5020 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥})) | 
| 6 | df-coll 44275 | . 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) | |
| 7 | df-coll 44275 | . 2 ⊢ (𝐺 Coll 𝐵) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥}) | |
| 8 | 5, 6, 7 | 3eqtr4g 2801 | 1 ⊢ (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 {csn 4625 ∪ ciun 4990 “ cima 5687 Scott cscott 44259 Coll ccoll 44274 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-iun 4992 df-br 5143 df-opab 5205 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-scott 44260 df-coll 44275 | 
| This theorem is referenced by: colleq1 44278 colleq2 44279 | 
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