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Mathbox for Rohan Ridenour |
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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 6016 | . . . 4 ⊢ (𝜑 → (𝐹 “ {𝑥}) = (𝐺 “ {𝑥})) |
4 | 3 | scotteqd 42609 | . . 3 ⊢ (𝜑 → Scott (𝐹 “ {𝑥}) = Scott (𝐺 “ {𝑥})) |
5 | 1, 4 | iuneq12d 4986 | . 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥})) |
6 | df-coll 42623 | . 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) | |
7 | df-coll 42623 | . 2 ⊢ (𝐺 Coll 𝐵) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥}) | |
8 | 5, 6, 7 | 3eqtr4g 2798 | 1 ⊢ (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 {csn 4590 ∪ ciun 4958 “ cima 5640 Scott cscott 42607 Coll ccoll 42622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-iun 4960 df-br 5110 df-opab 5172 df-cnv 5645 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-scott 42608 df-coll 42623 |
This theorem is referenced by: colleq1 42626 colleq2 42627 |
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