Theorem colleq12d 44244

Description: Equality theorem for the collection operation. (Contributed by Rohan Ridenour, 11-Aug-2023.)

Hypotheses

Ref	Expression
colleq12d.1	⊢ (𝜑 → 𝐹 = 𝐺)
colleq12d.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
colleq12d	⊢ (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵))

Proof of Theorem colleq12d

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		colleq12d.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		colleq12d.1	. . . . 5 ⊢ (𝜑 → 𝐹 = 𝐺)
3	2	imaeq1d 6051	. . . 4 ⊢ (𝜑 → (𝐹 “ {𝑥}) = (𝐺 “ {𝑥}))
4	3	scotteqd 44228	. . 3 ⊢ (𝜑 → Scott (𝐹 “ {𝑥}) = Scott (𝐺 “ {𝑥}))
5	1, 4	iuneq12d 5002	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥}))
6		df-coll 44242	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
7		df-coll 44242	. 2 ⊢ (𝐺 Coll 𝐵) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥})
8	5, 6, 7	3eqtr4g 2796	1 ⊢ (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵))

Syntax hints:   → wi 4   = wceq 1540  {csn 4606  ∪ ciun 4972   “ cima 5662  Scott cscott 44226   Coll ccoll 44241

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-iun 4974  df-br 5125  df-opab 5187  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-scott 44227  df-coll 44242

This theorem is referenced by:  colleq1  44245  colleq2  44246