Theorem colleq12d 40961

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 5895	. . . 4 ⊢ (𝜑 → (𝐹 “ {𝑥}) = (𝐺 “ {𝑥}))
4	3	scotteqd 40945	. . 3 ⊢ (𝜑 → Scott (𝐹 “ {𝑥}) = Scott (𝐺 “ {𝑥}))
5	1, 4	iuneq12d 4909	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥}) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥}))
6		df-coll 40959	. 2 ⊢ (𝐹 Coll 𝐴) = ∪ 𝑥 ∈ 𝐴 Scott (𝐹 “ {𝑥})
7		df-coll 40959	. 2 ⊢ (𝐺 Coll 𝐵) = ∪ 𝑥 ∈ 𝐵 Scott (𝐺 “ {𝑥})
8	5, 6, 7	3eqtr4g 2858	1 ⊢ (𝜑 → (𝐹 Coll 𝐴) = (𝐺 Coll 𝐵))

Syntax hints:   → wi 4   = wceq 1538  {csn 4525  ∪ ciun 4881   “ cima 5522  Scott cscott 40943   Coll ccoll 40958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-iun 4883  df-br 5031  df-opab 5093  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-scott 40944  df-coll 40959

This theorem is referenced by:  colleq1  40962  colleq2  40963