Theorem fnwe2val 43066

Description: Lemma for fnwe2 43070. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Hypotheses

Ref	Expression
fnwe2.su	⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)
fnwe2.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}

Assertion

Ref	Expression
fnwe2val	⊢ (𝑎𝑇𝑏 ↔ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))

Distinct variable groups:   𝑦,𝑈,𝑧,𝑎,𝑏   𝑥,𝑆,𝑦,𝑎,𝑏   𝑥,𝑅,𝑦,𝑎,𝑏   𝑥,𝑧,𝐹,𝑦,𝑎,𝑏   𝑇,𝑎,𝑏

Allowed substitution hints:   𝑅(𝑧)   𝑆(𝑧)   𝑇(𝑥,𝑦,𝑧)   𝑈(𝑥)

Proof of Theorem fnwe2val

Step	Hyp	Ref	Expression
1		vex 3483	. 2 ⊢ 𝑎 ∈ V
2		vex 3483	. 2 ⊢ 𝑏 ∈ V
3		fveq2 6905	. . . 4 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
4		fveq2 6905	. . . 4 ⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏))
5	3, 4	breqan12d 5158	. . 3 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ↔ (𝐹‘𝑎)𝑅(𝐹‘𝑏)))
6	3, 4	eqeqan12d 2750	. . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
7		simpl 482	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑎)
8		fvex 6918	. . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V
9		fnwe2.su	. . . . . . . 8 ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)
10	8, 9	csbie 3933	. . . . . . 7 ⊢ ⦋(𝐹‘𝑥) / 𝑧⦌𝑆 = 𝑈
11	3	csbeq1d 3902	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ⦋(𝐹‘𝑥) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑎) / 𝑧⦌𝑆)
12	10, 11	eqtr3id 2790	. . . . . 6 ⊢ (𝑥 = 𝑎 → 𝑈 = ⦋(𝐹‘𝑎) / 𝑧⦌𝑆)
13	12	adantr 480	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑈 = ⦋(𝐹‘𝑎) / 𝑧⦌𝑆)
14		simpr 484	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝑦 = 𝑏)
15	7, 13, 14	breq123d 5156	. . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥𝑈𝑦 ↔ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏))
16	6, 15	anbi12d 632	. . 3 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦) ↔ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))
17	5, 16	orbi12d 918	. 2 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦)) ↔ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏))))
18		fnwe2.t	. 2 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦))}
19	1, 2, 17, 18	braba 5541	1 ⊢ (𝑎𝑇𝑏 ↔ ((𝐹‘𝑎)𝑅(𝐹‘𝑏) ∨ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1539  ⦋csb 3898   class class class wbr 5142  {copab 5204  ‘cfv 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-iota 6513  df-fv 6568

This theorem is referenced by:  fnwe2lem2  43068  fnwe2lem3  43069