Theorem colleq12d 44834

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 6050	. . . 4 ⊢ (𝜑 → (𝐹 “ {𝑥}) = (𝐺 “ {𝑥}))
4	3	scotteqd 44818	. . 3 ⊢ (𝜑 → Scott (𝐹 “ {𝑥}) = Scott (𝐺 “ {𝑥}))
5	1, 4	iuneq12d 4981	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥}))
6		df-coll 44832	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
7		df-coll 44832	. 2 ⊢ (𝐺 Coll 𝐵) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥})
8	5, 6, 7	3eqtr4g 2824	1 ⊢ (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵))

Syntax hints:   → wi 4   = wceq 1562  {csn 4584  ∪ ciun 4951   “ cima 5652  Scott cscott 44816   Coll ccoll 44831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-iun 4953  df-br 5103  df-opab 5165  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-scott 44817  df-coll 44832

This theorem is referenced by:  colleq1  44835  colleq2  44836