Theorem fnwe2val 43006

Description: Lemma for fnwe2 43010. 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 3492	. 2 ⊢ 𝑎 ∈ V
2		vex 3492	. 2 ⊢ 𝑏 ∈ V
3		fveq2 6920	. . . 4 ⊢ (𝑥 = 𝑎 → (𝐹‘𝑥) = (𝐹‘𝑎))
4		fveq2 6920	. . . 4 ⊢ (𝑦 = 𝑏 → (𝐹‘𝑦) = (𝐹‘𝑏))
5	3, 4	breqan12d 5182	. . 3 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ↔ (𝐹‘𝑎)𝑅(𝐹‘𝑏)))
6	3, 4	eqeqan12d 2754	. . . 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 6933	. . . . . . . 8 ⊢ (𝐹‘𝑥) ∈ V
9		fnwe2.su	. . . . . . . 8 ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)
10	8, 9	csbie 3957	. . . . . . 7 ⊢ ⦋(𝐹‘𝑥) / 𝑧⦌𝑆 = 𝑈
11	3	csbeq1d 3925	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ⦋(𝐹‘𝑥) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑎) / 𝑧⦌𝑆)
12	10, 11	eqtr3id 2794	. . . . . 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 5180	. . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑥𝑈𝑦 ↔ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏))
16	6, 15	anbi12d 631	. . 3 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑈𝑦) ↔ ((𝐹‘𝑎) = (𝐹‘𝑏) ∧ 𝑎⦋(𝐹‘𝑎) / 𝑧⦌𝑆𝑏)))
17	5, 16	orbi12d 917	. 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 5556	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 846   = wceq 1537  ⦋csb 3921   class class class wbr 5166  {copab 5228  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-iota 6525  df-fv 6581

This theorem is referenced by:  fnwe2lem2  43008  fnwe2lem3  43009