Theorem colleq12d 45085

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 6059	. . . 4 ⊢ (𝜑 → (𝐹 “ {𝑥}) = (𝐺 “ {𝑥}))
4	3	scotteqd 9873	. . 3 ⊢ (𝜑 → Scott (𝐹 “ {𝑥}) = Scott (𝐺 “ {𝑥}))
5	1, 4	iuneq12d 4984	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥}))
6		df-coll 45083	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
7		df-coll 45083	. 2 ⊢ (𝐺 Coll 𝐵) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥})
8	5, 6, 7	3eqtr4g 2822	1 ⊢ (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  {csn 4587  ∪ ciun 4954   “ cima 5662  Scott cscott 9871   Coll ccoll 45082

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-iun 4956  df-br 5108  df-opab 5172  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-scott 9872  df-coll 45083

This theorem is used by:  colleq1  45086  colleq2  45087