Theorem fnwe2lem1 43039

Description: Lemma for fnwe2 43042. Substitution in well-ordering hypothesis. (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))}
fnwe2.s	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})

Assertion

Ref	Expression
fnwe2lem1	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})

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

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

Proof of Theorem fnwe2lem1

Step	Hyp	Ref	Expression
1		fnwe2.s	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
2	1	ralrimiva 3125	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
3		fveq2 6858	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝐹‘𝑎) = (𝐹‘𝑥))
4	3	csbeq1d 3866	. . . . . 6 ⊢ (𝑎 = 𝑥 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = ⦋(𝐹‘𝑥) / 𝑧⦌𝑆)
5		fvex 6871	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
6		fnwe2.su	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → 𝑆 = 𝑈)
7	5, 6	csbie 3897	. . . . . 6 ⊢ ⦋(𝐹‘𝑥) / 𝑧⦌𝑆 = 𝑈
8	4, 7	eqtrdi 2780	. . . . 5 ⊢ (𝑎 = 𝑥 → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 = 𝑈)
9	3	eqeq2d 2740	. . . . . 6 ⊢ (𝑎 = 𝑥 → ((𝐹‘𝑦) = (𝐹‘𝑎) ↔ (𝐹‘𝑦) = (𝐹‘𝑥)))
10	9	rabbidv 3413	. . . . 5 ⊢ (𝑎 = 𝑥 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
11	8, 10	weeq12d 5627	. . . 4 ⊢ (𝑎 = 𝑥 → (⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} ↔ 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)}))
12	11	cbvralvw 3215	. . 3 ⊢ (∀𝑎 ∈ 𝐴 ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)} ↔ ∀𝑥 ∈ 𝐴 𝑈 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑥)})
13	2, 12	sylibr 234	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})
14	13	r19.21bi 3229	1 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⦋(𝐹‘𝑎) / 𝑧⦌𝑆 We {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) = (𝐹‘𝑎)})

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  ⦋csb 3862   class class class wbr 5107  {copab 5169   We wwe 5590  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-iota 6464  df-fv 6519

This theorem is referenced by:  fnwe2lem2  43040  fnwe2lem3  43041