Theorem cdlemk41 41584

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 2932	. 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 6882	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑅‘𝑔) = (𝑅‘𝐺))
4	3	oveq2d 7427	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑃 ∨ (𝑅‘𝑔)) = (𝑃 ∨ (𝑅‘𝐺)))
5		coeq1 5844	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔 ∘ ◡𝑏) = (𝐺 ∘ ◡𝑏))
6	5	fveq2d 6886	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑅‘(𝑔 ∘ ◡𝑏)) = (𝑅‘(𝐺 ∘ ◡𝑏)))
7	6	oveq2d 7427	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏))) = (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏))))
8	4, 7	oveq12d 7429	. . 3 ⊢ (𝑔 = 𝐺 → ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏)))))
9	2, 8	eqtrid 2816	. 2 ⊢ (𝑔 = 𝐺 → 𝑌 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏)))))
10	1, 9	csbiegf 3894	1 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑌 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏)))))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ⦋csb 3861  ^◡ccnv 5661   ∘ ccom 5666  ‘cfv 6537  (class class class)co 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-co 5671  df-iota 6493  df-fv 6545  df-ov 7414

This theorem is referenced by:  cdlemkid2  41588  cdlemkfid3N  41589  cdlemky  41590  cdlemk42yN  41608