Theorem cdlemk41 41029

Description: Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 19-Jul-2013.)

Hypothesis

Ref	Expression
cdlemk41.y	⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏))))

Assertion

Ref	Expression
cdlemk41	⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑌 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏)))))

Distinct variable groups:   ∧ ,𝑔   ∨ ,𝑔   𝑔,𝐺   𝑃,𝑔   𝑅,𝑔   𝑇,𝑔   𝑔,𝑍   𝑔,𝑏

Allowed substitution hints:   𝑃(𝑏)   𝑅(𝑏)   𝑇(𝑏)   𝐺(𝑏)   ∨ (𝑏)   ∧ (𝑏)   𝑌(𝑔,𝑏)   𝑍(𝑏)

Proof of Theorem cdlemk41

Step	Hyp	Ref	Expression
1		nfcvd 2895	. 2 ⊢ (𝐺 ∈ 𝑇 → Ⅎ𝑔((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏)))))
2		cdlemk41.y	. . 3 ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏))))
3		fveq2 6822	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑅‘𝑔) = (𝑅‘𝐺))
4	3	oveq2d 7362	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑃 ∨ (𝑅‘𝑔)) = (𝑃 ∨ (𝑅‘𝐺)))
5		coeq1 5796	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔 ∘ ◡𝑏) = (𝐺 ∘ ◡𝑏))
6	5	fveq2d 6826	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑅‘(𝑔 ∘ ◡𝑏)) = (𝑅‘(𝐺 ∘ ◡𝑏)))
7	6	oveq2d 7362	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏))) = (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏))))
8	4, 7	oveq12d 7364	. . 3 ⊢ (𝑔 = 𝐺 → ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏)))))
9	2, 8	eqtrid 2778	. 2 ⊢ (𝑔 = 𝐺 → 𝑌 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏)))))
10	1, 9	csbiegf 3878	1 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑌 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏)))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ⦋csb 3845  ^◡ccnv 5613   ∘ ccom 5618  ‘cfv 6481  (class class class)co 7346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-co 5623  df-iota 6437  df-fv 6489  df-ov 7349

This theorem is referenced by:  cdlemkid2  41033  cdlemkfid3N  41034  cdlemky  41035  cdlemk42yN  41053