Theorem cdlemk41 42117

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 2926	. 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 6893	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑅‘𝑔) = (𝑅‘𝐺))
4	3	oveq2d 7443	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑃 ∨ (𝑅‘𝑔)) = (𝑃 ∨ (𝑅‘𝐺)))
5		coeq1 5839	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔 ∘ ◡𝑏) = (𝐺 ∘ ◡𝑏))
6	5	fveq2d 6897	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑅‘(𝑔 ∘ ◡𝑏)) = (𝑅‘(𝐺 ∘ ◡𝑏)))
7	6	oveq2d 7443	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏))) = (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏))))
8	4, 7	oveq12d 7445	. . 3 ⊢ (𝑔 = 𝐺 → ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏)))))
9	2, 8	eqtrid 2810	. 2 ⊢ (𝑔 = 𝐺 → 𝑌 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏)))))
10	1, 9	csbiegf 3883	1 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑌 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏)))))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  ⦋csb 3850  ^◡ccnv 5653   ∘ ccom 5658  ‘cfv 6546  (class class class)co 7427

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-co 5663  df-iota 6500  df-fv 6554  df-ov 7430

This theorem is used by:  cdlemkid2  42121  cdlemkfid3N  42122  cdlemky  42123  cdlemk42yN  42141