Theorem colleq12d 44712

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 6018	. . . 4 ⊢ (𝜑 → (𝐹 “ {𝑥}) = (𝐺 “ {𝑥}))
4	3	scotteqd 44696	. . 3 ⊢ (𝜑 → Scott (𝐹 “ {𝑥}) = Scott (𝐺 “ {𝑥}))
5	1, 4	iuneq12d 4954	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥}))
6		df-coll 44710	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
7		df-coll 44710	. 2 ⊢ (𝐺 Coll 𝐵) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥})
8	5, 6, 7	3eqtr4g 2801	1 ⊢ (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵))

Syntax hints:   → wi 4   = wceq 1548  {csn 4558  ∪ ciun 4924   “ cima 5624  Scott cscott 44694   Coll ccoll 44709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-iun 4926  df-br 5076  df-opab 5138  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-scott 44695  df-coll 44710

This theorem is referenced by:  colleq1  44713  colleq2  44714